


Plane and Spherical 



{\IGONOMETI\Y 



Lyman and Goddard 



With Tables 



LIBRARY OF CONGRESS. 



QA53* 

Chap.__ ' Copyright No. 

Shelf_,__i_lE 



UNITED STATES OF AMERICA. 



PLANE AND SPHERICAL 
TRIGONOMETRY 



B V 

ELMER A. LYMAN 

MICHIGAN STATE NORMAL COLLEGE 
AND 

EDWIN C. GODDARD 

UNIVERSITY OF MICHIGAN 



>XK' 



ALLYN AND BACON 

Boston arto Chicago 



i 



67775 



OCT 29 1900 

Cofynghl entry 

SECOND COPY. 
GftGW DMS10N, 

OCT 30 I9UU 



1 1* 



COPYEIGHT, 189 9, 19 00, 
BY ELMER A. LYMAN 
AND EDWIN C. GODDARD. 



Nortoooti ^reas 

J. S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 



It 






t 



PREFACE. 



Many American text-books on trigonometry treat the solution 
of triangles quite fully; English text-books elaborate analytical 
trigonometry; but no book available seems to meet both needs 
adequately. To do that is the first aim of the present work, in 
the preparation of which nearly everything has been worked out 
and tested by the authors in their classes. 

The work entered upon,, other features demanded attention 
For some unaccountable reason nearly all books, in proving the 
formulae for functions of a ± ft treat the same line as both posi- 
tive and negative, thus vitiating the proof; and proofs given for 
acute angles are (without further discussion) supposed to apply 
to all angles, or it is suggested that the student can draw other 
figures and show that the formulae hold in all cases. As a 
matter of fact the average student cannot show anything of the 
kind; and if he could, the proof would still apply only to combi- 
nations of conditions the same as those in the figures actually 
drawn. These difficulties are avoided by so wording the proofs 
that the language applies to figures involving any angles, and to 
avoid drawing the indefinite number of figures necessary fully 
to establish the formulae geometrically, the general case is proved 
algebraically (see page 58). 

Inverse functions are introduced early, and used constantly 
Wherever computations are introduced they are made by means 
of logarithms. The average student, using logarithms for a short 
time and only at the end of the subject, straightway forgets what 
manner of things they are. It is hoped, by dint of much prac- 
tice, extended over as long a time as possible, to give the student 
a command of logarithms that will stay. The fundamental for- 
mulae of trigonometry must be memorized. There is no substi- 
tute for this. For this purpose oral work is introduced, and 
there are frequent lists of review problems involving all prin- 
ciples and formulae previously developed. These lists serve the 

iii 



iv PREFACE. 

further purpose of throwing the student on his own resources, 
and compelling him to find in the problem itself, and not in any 
model solution, the key to its solution, thus developing power, 
instead of ability to imitate. To the same end, in the solution 
of triangles, divisions and subdivisions into cases are abandoned, 
and the student is thrown on his own judgment to determine 
which of the three possible sets of formulae will lead to the solu- 
tions with the data given. Long experience justifies this as 
clearer and simpler. The use of checks is insisted upon in all 
computations. 

For the usual course in plane trigonometry Chapters I-VII, 
omitting Arts. 26, 27, contain enough. Articles marked * (as 
Art. # 26) may be omitted unless the teacher finds time for them 
without neglecting the rest of the work. Classes that can accom- 
plish more will find a most interesting field opened in the other 
chapters. More problems are provided than any student is ex- 
pected to solve, in order that different selections may be assigned 
to different students, or to classes in different years. Do not 
assign ivork too fast. Make sure the student has memorized and 
can use each preceding formula, before taking up new ones. 

No complete acknowledgment of help received could here be 
made. The authors are under obligation to many for general 
hints, and to several who, after going over the proof with care, 
have given valuable suggestions. The standard works of Levett 
and Davison, Hobson, Henrici and Treutlein, and others have 
been freely consulted, and while many of the problems have been 
prepared Vv the authors in their class-room work, they have not 
hesitated to take, from such standard collections as writers gen- 
erally have drawn upon, any problems that seemed better adapted 
than others to the work. Quality has not been knowingly sacri- 
ficed to originality. Corrections and suggestions will be gladly 
received at any time. 

E. A. L., Ypsilanti. 
E. C. G., Ann Arbor. 

October, 1900. 



CONTENTS. 

Chapter I. Angles — Measurement of Angles. 

PAGE 

Angles ; magnitude of angles 1 

Rectangular axes ; direction 2 

Measurement; sexagesimal and circular systems of measurement; 

the radian 3 

Examples 6 

Chapter II. The Trigonometric Functions. 

Function defined 8 

The trigonometric functions . 9 

Fundamental relations .11 

Examples . . 14 

Functions of 0°, 30°, 45°, 60°, 90° 15 

Examples 18 

Variations in the trigonometric functions 19 

Graphic representation of functions 22 

Examples 27 

Chapter III. Functions of any Angle — Inverse 
Functions. 

Relations of functions of - $, 90° ± 0, 180° ± $, 270° ± to the 

functions of 29 

Inverse functions .35 

Examples 36 

Review 38 

Chapter IV. Computation Tables. 

Natural functions 40 

Logarithms 40 

Laws of logarithms 42 

Use of tables 45 

Cologarithms 49 

Examples 50 

v 



vi CONTENTS. 

Chapter V. Applications. 

PAGE 

Measurements of heights and distances 51 

Common problems in measurement . 52 

Examples . . . .54 

Chapter VI. General Formulae — Trigonometric 
Equations and Identities. 

Sine, cosine, tangent of a ± (3 56 

Examples 59 

Sin ± sin cf>, ± cos <f> 61 

Examples 62 

Functions of the double angle . . 63 

Functions of the half angle . . . . . . . .64 

Examples . . . 64 

Trigonometric equations and identities ...... 66 

Method of attack . . * . . 66 

Examples . . . .67 

Simultaneous trigonometric equations . . . . . .69 

Examples . . . 70 

Chapter VII. Triangles. 

Laws of sines, tangents, and cosines 72 

Area of the triangle 76 

Solution of triangles . . .76 

Ambiguous case 78 

Model solutions . .80 

Examples .83 

Applications 84 

Review 86 

Chapter VIII. Miscellaneous. 

Incircle, circumcircle, escribed circle 92 

Orthocentre, centroid, medians 94 

Examples .96 

Chapter IX. Series. 

Exponential series 97 

Logarithmic series .99 

Computation of logarithms 100 

De Moivre's theorem . . 103 

Computation of natural functions 104 

Hyperbolic functions 109 

Examples 110 



CONTENTS. vii 
Chapter X. Spherical Trigonometry. 

PAGE 

Spherical triangles 112 

General formulae 114 

Right spherical triangles 123 

Area of spherical triangles 125 

Examples 128 

Chapter XL Solution of Spherical Triangles. 

General principles 129 

Formulae for solution 130 

Model solutions 131 

Ambiguous cases 132 

Right triangles . . • 134 

Species 135 

Examples 137 

Applications to Geodesy and Astronomy 138 



.PLANE TRIGONOMETRY. 

CHAPTER I. 

ANGLES — MEASUREMENT OF ANGLES. 

1. Angles. It is difficult, if not impossible, to define an 
angle. This difficulty may be avoided by telling how it 
is formed. If a line revolve about one of its points, an angle 
is generated, the magnitude of the angle depending on the 
amount of the rotation. 

Thus, if one side of the angle 6, as OR, be originally in 
the position OX, and be revolved about the point to the 
position in the figure, the 
angle XOR is generated. 
OX is called the initial line, 
and any position of OR the 
terminal line of the angle 
formed. The angle is 
considered positive if gener- 
ated by a counter-clockwise 
rotation of OR, and hence negative if generated by a clockwise 
rotation. The magnitude of 6 depends on the amount of 
rotation of OR, and since the amount of such rotation may 
be unlimited, there is no limit to the possible magnitude of 
angles, for, evidently, the revolving line may reach the posi- 
tion OR by rotation through an acute angle 6, and, likewise, 
by rotation through once, twice, ••♦, n times 360°, plus the 
acute angle 6. So that XOR may mean the acute angle 
6, 6 + 360°, 6 + 720°, •••, 6 + n • 360°. 

l 




R. 



y o 

Fig. 1. 



2 PLANE TRIGONOMETRY. 

In reading an angle, read first the initial line, then the 
terminal line. Thus in the figure the acute angle XOR, or 
xr, is a positive angle, and ROX, or rx, an equal negative 
angle. 

Ex. 1. Show that if the initial lines for \, f, - 2 ^, — f , right angles are 
the same, the terminal lines may coincide. 

2. Name four other angles having the same initial and terminal lines 
as £ of a right angle ; as f of a right angle ; as f of a right angle. 

2. Rectangular axes. Any plane surface may be divided 
by two perpendicular straight lines XX' and YY' into four 

portions, or quadrants. 

XX' is known as the x-axis, 
YY' as the y-axis, and the two 
together are called axes of refer- 

X- Lz X enee. Their intersection is the 

origin, and the four portions of 
the plane surface, XOY, YOX', 
Y > X'OY', Y'OX, are called respec- 

FlG 2. tively the first, second, third, and 

fourth quadrants. The position of 
any point in the plane is determined when we know its dis- 
tances and directions from the axes. 

3. Any direction may be considered positive. Then the 
opposite direction must be negative. Thus, if AB represents 
any positive line, BA is an equal nega- 
tive line. Mathematicians usually 

consider lines measured in the same direction as OX or OY 
(Fig. 2) as positive. Then lines measured in the same direc- 
tion as OX' or OY' must be negative. 

The distance of any point from the ?/-axis is called the 
abscissa, its distance from the #-axis the ordinate, of that 
point ; the two together are the coordinates of the point, 
usually denoted by the letters x and y respectively, and 
written (x, y). 



ANGLES — MEASUREMENT. 3 

When taken with their proper signs, the coordinates define completely 
the position of the point. Thus, if the point P is + a units from YY', 
and + b units from XX', any convenient 
unit of length being chosen, the position of 
P is known. For we have only to measure 
a distance ON equal to a units along OX, 
and then from N measure a distance b 
units parallel to OY, and we arrive at the 
position of the point P, (a, b). In like 
manner we may locate P', ( — a, b), in the 
second quadrant, P", (—a, — 6), in the 
third quadrant, and P'", (a, —6), in 
the fourth quadrant. 







Y 






p ' 




P 









N 


p" 




P'" 






Y' 





Fig. 3. 



Ex. Locate (2, 
(ro, n). 



2); (0,0); (-8, -7); (0. 5); (-2, 0); (2, 2) 



R R 



R X X R 



4. If OX is the initial line, 6 is said to be an angle of the 
first, second, third, or fourth quadrant, according as its ter- 
minal line is in the first, second, third, or fonrth quadrant. 
It is clear that as OR rotates its quality is in no wa} T affected, 
and hence it is in all positions considered positive, and its ex- 
tension through 0, OR', negative. 

The student should notice that the initial line may take any position 
and revolve in either direction. While it is customary to consider the 
counter-clockwise rotation as forming a positive angle, yet the condi- 
tions of a figure may be such 
that a positive angle may be 
X' V generated by a clockwise rota- 

rO AO 

"■#' ■■'' \ A ^ on " Thus the angle XOR in 
x' r'r \x each figure may be traced as 
a positive angle by revolving 
the initial line OX to the posi- 
tion OR. Xo confusion can result if the fact is clear that when an 
angle is read XOR, OX is considered a positive line revolving to the 
position OR. OX' and OR' then are negative lines in whatever direc- 
tions drawn. These conceptions are mere matters of agreement, and the 
agreement may be determined in a particular case by the conditions of 
the problem quite as well as by such general agreements of mathema- 
ticians as those referred to in Arts. 3 and 4 above. 



Fig. 4. 



5. Measurement. All measurements are made in terms 
of some fixed standard adopted as a unit. This unit must 



4 PLANE TRIGONOMETRY. 

be of the same kind as the quantity measured. Thus, length 
is measured in terms of a unit length, surface in terms of a 
unit surface, weight in terms of a unit weight, value in terms 
of a unit value, an angle in terms of a unit angle. 

The measure of a given quantity is the number of times it 
contains the unit selected. 

Thus the area of a given surface in square feet is the 
number of times it contains the unit surface 1 sq. ft. ; the 
length of a road in miles, the number of times it contains 
the unit length 1 mi. ; the weight of a cargo of iron ore in 
tons, the number of times it contains the unit weight 1 ton ; 
the value of an estate, the number of times it contains the 
unit value $1. 

The same quantity may have different measures, according 
to the unit chosen. So the measure of 80 acres, when the 
unit surface is 1 acre, is 80, when the unit surface is 1 sq. rd., 
is 12,800, when the unit surface is 1 sq. yd., is 387,200. 
What is its measure in square feet ? 

6. The essentials of a good unit of measure are : 

1. That it be invariable, i.e. under all conditions bearing 
the same ratio to equal magnitudes. 

2. That it be convenient for practical or theoretical pur- 
poses. 

3. That it be of the same kind as the quantity measured. 

7. Two systems of measuring angles are in use, the sexa- 
gesimal and the circular. 

The sexagesimal system is used in most practical applica- 
tions. The right angle, the unit of measure in geometry, 
though it is invariable, as a measure is too large for con- 
venience. Accordingly it is divided into 90 equal parts, 
called degrees. The degree is divided into 60 minutes, and 
the minute into 60 seconds. Degrees, minutes, seconds, are 
indicated by the marks ° ' ", as 36° 20' 15". 

The division of a right angle into hundredths, with subdivisions into* 
hundredths, would be more convenient. The French have proposed such 



MEASUREMENT OF ANGLES. 



a centesimal system, dividing the right angle into 100 grades, the grade 
into 100 minutes, and the minute into 100 seconds, marked R x XN , as 50s 
70 x 28". The great labor involved in changing mathematical tables, 
instruments, and records of observation to the new system has prevented 
its adoption. 

8. The circular system is important in theoretical con- 
siderations. It is based on the fact that for a given angle 
the ratio of the length of its arc to the length of the radius 
of that arc is constant, i.e. for a fixed 
angle the ratio arc : radius is the same 
no matter what the length of the > & 

radius. In the figure, for the angle 6, 

OA _ OB _ OO = 




AA' BB< 



That this ratio of arc to radius for a fixed angle is constant 
follows from the established geometrical principles : 

1. The circumference of any circle is 2 ir times its radius. 

2. Angles at the centre are in the same ratio as their arcs. 

The Radian. It follows that an angle whose arc is equal 
in length to the radius is a constant angle for all circles, 
since in four right angles, or the perigon, there are always 
2tt such angles. This constant angle, 
whose arc is equal in length to the radius, 
is taken as the unit angle of circular 
measure, and is called the radian. From 
the definition we have 

4 right angles = 360° = 2 ir radians, 

2 right angles = 180° = it radians, 

7T 




Fig. 



1 right angle = 90° = — radians 



7T is a numerical quantity, 3.14159+, and not an angle. When we 
speak of 180° as it, 90° as ^, etc., we always mean -k radians, - radians, etc. 



6 PLANE TRIGONOMETRY. 

9. To change from one system of measurement to the 
other we use the relation, 

2?r radians = 360°. 

1 80° 
.-. 1 radian = —=57°. 2958-; 

IT 

i.e. the radian is 57°. 3, approximately. 

Ex. 1. Express in radians 75° 30'. 

75° 30' = 75°.5; 1 radian = 57°.3. 

.♦. 75° 30' = — = 1.317 radians. 
57.3 

2. Express in degree measure 3.6 radians. 
1 radian = 57°. 3. 
.-. 3.6 radians = 3.6 x 57°.3 = 206° 16' 48". 

EXAMPLES. 

1. Construct, approximately, the following angles : 50°, — 20°, 90°, 
179°, - 135°, 400°, - 380°, 1140°, - radians, - radians, - - radians, 

q If) 

3 7r radians, - radians, — - radians. Of which quadrant is each 

angle? 4 5 

2. What is the measure of : 

(a) 4 of a right angle, when 30° is the unit of measure ? 

(b) an acre, when a square whose side is 10 rds. is the unit? 

(c) m miles, when y yards is the unit ? 

3. What is the unit of measure, when the measure of 2\ miles is 50? 

4. The Michigan Central R.R. is 535 miles long, and the Ann Arbor 
R.R. is 292 miles long. Express the length of the first in terms of the 
second as a unit. 

5. What will be the measure of the radian when the right angle is 
taken for the unit? Of the right angle when the radian is the unit? 

6. In which quadrant is 45°? 10°? -60°? 145°? 1145°? -725°? 
Express each in right angles ; in radians. 

7. Express in sexagesimal measure 

£, -£, 1, 6.28, 1, Ii; - i^ radians. 
3 12 7T 3 3 ' 



EXAMPLES. 7 

8. Express in each system an interior angle of a regular hexagon ; 
an exterior angle. 

9. Find the distance in miles between two places on the earth's 
equator which are 11° 15( apart. (The earth's radius is about 3963 miles.) 

10. Find the length of an arc which subtends an angle of 4 radians 
at the centre of a circle of radius 12 ft. 3 in. 

11. An arc 15 yds. long contains 3 radians. Find the radius of the 
circle. 

12. Show that the hour and minute hands of a watch turn through 
angles of 30' and 6° respectively per minute ; also find in degrees and in 
radians the angle turned through by the minute hand in 3 hrs. 20 mins. 

13. Find the number of seconds in an arc of 1 mile on the equator ; 
also the length in miles of an arc of 1' (1 knot). 

14. Find to three decimal places the radius of a circle in which the 
arc of 71° 36' 3".6 is 15 in. long. 

15. Find the ratio of - to 5°. 

6 

16. What is the shortest distance measured on the earth's surface 
from the equator to Ann Arbor, latitude + 42° 16' 48" ? 

17. The difference of two angles is 10°, and the circular measure of 
their sum is 2. Find the circular measure of each angle. 

18. A water wheel of radius 6 ft. makes 30 revolutions per minute. 
Find the number of miles per hour travelled by a point on the rim. 



CHAPTER II. 

THE TRIGONOMETRIC FUNCTIONS. 

10. Trigonometry, as the word indicates, was originally 
concerned with the measurement of triangles. It now 
includes the analytical treatment of certain functions of 
angles, as well as the solution of triangles by means of cer- 
tain relations between the functions of the angles of those 
triangles. 

11. Function. If one quantity depends upon another for 
its value, the first is called a function of the second. It 
always follows that the second quantity is also a function of 
the first ; and, in general, functions are so related that if one 
is constant the other is constant, and if either varies in value, 
the other varies. This relation may be extended to any 
number of mutually dependent quantities. 

Illustration. If a train moves at a rate of 30 miles per 
hour, the distance travelled is a function of the rate and 
time, the time is a function of the rate and distance, and the 
rate is a function of the time and distance. 

Again, the circumference of a circle is a function of the 
radius, and the radius of the circumference, for so long as 
either is constant the other is constant, and if either changes 
in value, the other changes, since circumference and radius 
are connected by the relation (7=2 irR. 

Once more, in the right triangle 
NOP, the ratio of any two sides is 
a function of the angle a, because 
all the right triangles of which a is 
one angle are similar, i.e. the ratio 
8 




THE TRIGONOMETRIC FUNCTIONS. 



9 



of two corresponding sides is constant so long as a is con- 
stant, and varies if a varies. 
Thus, the ratios 

NP N'P' N"P" 



and 



OP OP' 
ON ON' 



OP' 
ON 



, etc., 



NP N'P' N"P' 
depend on a for their values, i.e. are functions of a. 

12. The trigonometric functions. In trigonometry six 
functions of angles are usually employed, called the trigono- 
metric functions. 

By definition these functions are the six ratios betiveen the 
sides of the triangle of reference of the given angle. The 
triangle of reference is formed.bg drawing from some point in 
the initial line, or the initial line produced, a perpendicular to 
that line meeting the terminal line of the angle. 







Y 














P/ 


y R 






r> 




y 




~y' 




/&\ 













X 


N 


X 






Y' 










Let a be an angle of any quadrant. Each triangle of 
reference of a, NOP, is formed by drawing a perpendicular 
to OX, or OX produced, meeting the terminal line OB in P. 



10 PLANE TRIGONOMETRY. 

If a is greater than 360°, its triangle of reference would 
not differ from one of the above triangles. 

It is perhaps worthy of notice that the triangle of reference might be 
defined to be the triangle formed by drawing a perpendicular to either 
side of the angle, or that side produced, meet- 
ing the other side or the other side produced. 
In the figure, NOP is in all cases the triangle 
of reference of a. The principles of the fol- 



2? P /<a 




\ S O P N 

j ,.--'jy- lowing pages are the same no matter which 

s'-p of the triangles is considered the triangle of 

Fig. 9. reference. It will, however, be as well, and 

perhaps clearer, to use the triangle defined 
under Fig. 8, and we shall always draw the triangle as there described. 

13. The trigonometric functions of a (Fig. 8) are called 
the sine, cosine, tangent, cotangent, secant, and cosecant of a. 
These are abbreviated in writing to sin a, cos a, tan a, cot a, 
sec a, esc a, and are defined as follows : 

sin a = *? erp * = — 9 whence y = r sin a ; 
hyp. r * 

base x i 
cos a = c — = —9 whence x = r cos a ; 
hyp. r 

tan a = p . p * = — > whence y = x tan a ; 
base vc * 

C0t a = perp: = y ' wnence % = ycota; 

sec a = |-^- = —9 whence r = x sec a ; 
Dase oc 

esc a = — ^— = — 9 whence r = y esc a. 
perp. y * 

1 — cos a and 1 — sin a, called versed-sine a and coversed-sine a, respec- 
tively, are sometimes used. 

Ex. 1. Write the trigonometric functions of (3, NPO (Fig. 8), and 
compare with those of a above. 

The meaning of the prefix co in cosine, cotangent, and cosecant 
appears from the relations of Ex. 1. For the sine of an angle equals the 
cosine, i.e. the complement-sine, of the complement of that angle ; the tangent 



THE TRIGONOMETRIC FUNCTIONS. 



11 




of an angle equals the cotangent of its complementary angle, and the secant 
of an angle equals the cosecant of its complement- 
ary angle. 

2. Express each side of triangle ABC in 
terms of another side, and some function of an 
angle in all possible ways, as a = b tan A, etc. Fig. 10. 

14. Constancy of the trigonometric functions. It is impor- 
tant to notice why these ratios are functions of the angle, i.e. 
are the same for equal angles and different for unequal 
angles. This is shown by the principles of similar triangles. 




\ 




Fig. 11. 

In each figure show that in all possible triangles of refer- 
ence for a the ratios are the same, but in the triangles of 
reference for a and a', respectively, the ratios are different. 

The student must notice that sin a is a single symbol. It is the name 
of a number, or fraction, belonging to the angle a ; and if it be at any 
time convenient, we may denote sin a by a single letter, such as o, or x. 
Also, sin 2 a is an abbreviation for (sin a) 2 , i.e. for (sin a) x (sin a) . 
Such abbreviations are used because they are convenient. Lock, Ele- 
mentary Trigonometry. 

15. Fundamental relations. From the definitions of Art. 13 
the following reciprocal relations are apparent : 

1 



sm a — . 

csc a 


csc a = — > 

sin a 


cos a = — ?— , 

sec a 


1 

sec a — •> 

cos a 


tan a = —=—> 

cot a 


1 

cot a = 

tan a 


Also from the definitions, 




tana = siu S 
cos a 


, cos a 
cot a = — 



12 PLANE TRIGONOMETRY. 

From the right triangle NOP, page 9, 
y*> 4. x 2 = r 2 ; 

whence (1) ^ + ^ = 1, 



( 2 ) ^ + 1 



2 
+ 1 = 



( 3 > i +-2=4 

*/ 2 «/ 2 

From (1) sin 2 a + cos 2 a = 1 $ sin a=^/l — cos 2 a; cos a = ? 



(2) tan 2 a + 1 = sec 2 a ; taw a=-\Jsec 2 a— 1 ; sec«=? 

(3) l+cot 2 a = csc 2 a; co£ a= Vcsc 2 a— 1 ; esc « = ? 

^e foregoing definitions and fundamental relations are of 
the highest importance, and must be mastered at once. The 
student of trigonometry is helpless without perfect familiarity 
ivith them. 

These relations are true for all values of a, positive or negative, but 
the signs of the functions are not in all cases positive, as appears from 
the fact that in the triangles of reference in Fig. 8 x and y are sometimes 
negative. The equations sin a =± Vl — cos 2 a, tan a = ± Vsec 2 a — 1, 
cot a = ± Vcsc 2 a — 1, have the double sign ±. Which sign is to be used 
in a given case depends on the quadrant in which a lies. 

16. The relations of Art. 15 enable us to express any 
function in terms of any other, or when one function is 
given, to find all the others. 

Ex. 1. To express the other functions in terms of tangent : 
sin « = ^_ = 1 = tan a : cot«= 1 



esc a Vl + cot 2 a Vl + tan 2 a tan a 

vTT 

Vl + tan* 2 a 



cosa = = — ; seca = Vl + tan 2 a; 

sec a Vl + tan 2 a 



tan a = tan a ; esc a = 



tan a 



THE TRIGONOMETRIC FUNCTIONS. 



13 



In like manner determine the relations to complete the following 
table : 



sin a 

cosce 
tan a 
cot a 
sec a 

CSCCJ 



sin a 



tan a 



tan a 



Vl + tan 2 a 
1 



VI + tan' 2 a 
tan a 

1 

tan a 

Vl + tan 2 a 

Vl + tan 2 a 



tan a 



cot « 



2. Given sin a = f ; find the other functions. 

cos a =V1 - i 9 e = i V7 ; tan a = — i- = — = f V7 ; 

iV7 V7 



cot a = — ^ = $ V7 ; sec a = - — _ 

fV7 iV7 V7 



*V7; csca=i = i 



t 3 



3. Given tan <f> -f cot <£ = 2 ; find sin <£ 

= 2, tan 

,\ sin <£ = 



tan <£ H — = 2, tan 2 <£ - 2 tan <£ + 1 = 0, tan <f> = 1. 

tan cf> 

tan <£ = ^ 



Vl + tan 2 <£ 

Or, expressing in terms of sine directly, ^H £ = 2, 

cos (£ sin <£ 

sin 2 <£ + cos 2 cfi = 2 sin <£ cos <£, sin 2 <£ — 2 sin <£ cos <£ + cos 2 <f> = 0; 
whence sin <£ — cos (£ = 0, sin <£ = cos <£. .*. sin <f> = |V2. 

4. Prove sec 4 a; — sec 2 x = tan 2 x + tan 4 a\ 

sec 4 a: — sec 2 z = sec 2 x (sec 2 x — 1) = (1 + tan 2 x) tan 2 x = tan 2 a: -f tan 4 x. 

5. Prove sin 6 ?/ -f cos 6 ?/ = 1 — 3 sin 2 y cos 2 y. 

sin 6 j/ + cos 6 y = (sin 2 y + cos 2 ?/) (sin 4 y — sin 2 ?/ cos 2 y + cos 4 #) 

= (sin 2 y + cos 2 #) 2 — 3 sin 2 y cos 2 y = 1 — 3 sin 2 ?/ cos 2 #. 



14 PLANE TRIGONOMETRY. 



6. 


x>„~™ tan z , cot z , n 

Prove h = sec z esc z + 1. 

1 — cot z 1 — tan z 




sin 2 cos z 
tan z cot z cos z sin z 




1 — cot z ' 1 — tan z , cos z + sin z 




sm z cos z 




sin 2 z cos 2 z 




cos z (sin z — cos z) ' sin z (cos z — sin z) 




sin 3 z — cos 3 z _ sin 2 z + sin z cos z -f cos 2 z 




sin z cos z (sin z — cos z) sin z cos z 




1 + sin z cos z 1 , .. , -, 




sm z cos z sin z cos z 



In solving problems like 3, 4, 5, and 6 above, it is usually safe, if no 
other step suggests itself, to express all other functions of one member 
in terms of sine and cosine. The resulting expression may then be re- 
duced by the principles of algebra to the expression in the other member 
of the equation. For further suggestions as to the solution of trigono- 
metric equations and identities see page 66. 

EXAMPLES. 

1. Find the values of all the functions of a, if sin cc = § ; if tan a = f ; 
if sec a = 2 ; if cos a = %V3 ; if cot a = \ ; if esc a = V'2. 

2. Compute the functions of each acute angle in the right triangles 

whose sides are: (1) 3, 4, 5; (2) 8, 15, 17; (3) 480, 31, 481; (4) a,b,c, 

/KN 2 xy x' 2 + V 2 

(5) *-, * , x + y. 

x — y x — y 

3. If cos a = T 8 7 , find the value of sill " + tanflg . 

cos a — cot a 

4. If 2 cos a = 2 — sin a, find tan a. 

5. If sec 2 a esc 2 a — 4 = 0, find cot a. 

6. Solve for sin fi in 13 sin fi + 5 cos 2 /3 = 11. 
Prove 

7. sin 4 <£ - cos 4 cf> = 1 - 2 cos 2 <£. 

8. (sin a + cos a) (sin cc — cos a) = 2 sin 2 « — 1. 

9. (sec a + tan cc) (sec « — tan a) = 1. 

10. cos 2 (sec 2 /? - 2 sin 2 /?) = cos 4 (3 + sin 4 0. 

COS V 



11. tan v + sect' = 
12. 



1 — sin v 

sin w 1 + cos w 



1 — cos w sin m> 
13. (sec + 1) (1 - cos $) = tan 2 cos ft 



FUNCTIONS OF CERTAIN ANGLES. 15 

14. sin 4 1 — sin 2 1 = cos 4 t — cos 2 t. 

15. J"L^ + !±il^ = sec*/? (csc /J + 1). 
1 - sin /tJ sin (3 

16. (tan ^4 -f cot^4) 2 = sec 2 .4 esc 2 ^4. 

17. sec 2 x — sin 2 x = tan 2 # + cos 2 x. 

In the triangle ABC, right angled at C, 

18. Given cos A = T 8 7 , BC = 45, find tan B, and t!5. 



?» 



Fig. 12. 



19. If cos A = "" ~~ , and AB = m 2 + n 2 , find A C and £C. 

m 2 + n 2 

20. If A C = m + n, BC = m — n, find sin A, cos B. 

21. In examples 18, 19, 20, above, prove sin 2 A + cos 2 A = 1 ; 
1 + tan 2 A = sec 2 A. 

17. Functions of certain angles. The trigonometric func- 
tions are numerical quantities which may be determined for 
any angle. In general these values are taken from tables 
prepared for the purpose, but the principles already studied 
enable us to calculate the functions of the following angles. 

18. Functions of 0°. If a be a very small angle, the 
value of y is very small, and 
decreases as a diminishes. 
Clearly, when a approaches 
0° as a limit, y likewise ap- 
proaches 0, and x approaches r, so that when a = 0°, 

y = 0, and x — r. 

= 00, 



1, 



= 00. 

Ill the figure of Art. 18, by diminishing a it is clear that we can make 
y as small as we please, and by making a small enough, we can make the 
value of y less than any assignable quantity, however small, so that sin a ap- 
proaches as a limit 0. This is what we mean when we say sin 0° = 0. 
In like manner, it is evident that, by sufficiently diminishing a we can 
make cot a greater than any assignable quantity. This we express by 
saying cot 0° = oo. 



sin 0° = £ = 0, 
r 


C0t 0° = — -r- 

tan 0° 


cos 0° = - = 1, 
r 


~ rr o° — 


" WU "cost) 


to o°=£ = o, 


«°V = ±TiS 



16 



PLANE TRIGONOMETRY. 



19. Functions of 30°. Let NOP be the triangle of refer- 
,# ence for an angle of 30°. Make 

triangle NOP' = NOP. Then 
POP' is an equilateral triangle 
(why?), and ON bisects PP'. 
Hence 

PP' = r== 2y. 




sin 30° 



y y 



2y 2 



Also x = Vr 2 — y 2 — V3 y 2 = y V3. 
esc 30° = 2, 



r 2y z 



tan 30° = 



y _ y 



yV3 V3 



sec 30° = f V3, 
W3, ^30°=V3. 



20. Functions of 45°. Let NOP be the triangle of refer- 
ence. If angle NOP = 45°, OPN= 45°. 




Then 



y = #, and r = Vx 2 + y 2 = V2 z 2 = jv2. 



sm 45° = v - 



:V2 



cos 45° = - = 



V2 



JV2, 
i-V2, 



tow 45° = y - = - = 1. 
a; a? 

Find <?o£ 45°, sec 45°, esc 45°. 



FUNCTIONS OF CERTAIN ANGLES. 



17 



21. Functions of 60°. The functions of 60° may be com- 
puted by means of the figure, or 
they may be written from the func- 
tions of the complement, or 30°. 
Let the student in both ways show 
that 

*m60°= JV3, cos60°=± 

tan 60° = VS. 

Compute also the other functions of 60°. 




22. Functions of 90 c 



Y 



xy- 



Fig. 16. 



. If a be an angle very near 90°, 
the value of x is very small, and de- 
creases as cc increases toward 90°. 
Clearly when a approaches 90° as a 
limit, x approaches 0, and y ap- 
proaches r, so that when 

a =90°, x=0, y = r. 

. •. sin 90° = 1, cos 90° = 0, tan 90° = oo . 



Compute the other functions. Also find the functions of 
90° from those of its complement, 0°. 

23. It is of great convenience to the student to remember 
the functions of these angles. They are easily found by 
recalling the relative values of the sides of the triangles of 
reference for the respective angles, or the values of the other 
functions may readily be computed by means of the funda- 
mental relations, if the values of the sine and cosine are 
remembered, as follows : 



a 


0° 


30° 


45° 


60° 


90° 


sine 
cosine 


Wo 


iVI 
4V3 


iV2 
W2 


iV3 
ivl 


iVo 



18 PLANE TRIGONOMETRY. 

ORAL WORK. 

1. Which is greater, sin 45° or \ sin 90° ? sin 60° or 2 sin 30° ? 

2. From the functions of 60°, find those of 30° ; from the functions of 
90°, those of 0°. Why are the functions of 45° equal to the co-functions 
of 45°? 

3. Given sin A =\, find cos A ; tan A. 

4. Show that sin BcscB = 1 ; cos C sec C = 1 ; cot x tan x = 1. 

5. Show that sec 2 - tan 2 = esc 2 6 - cot 2 d = sin 2 + cos 2 6. 

6. Show that tan 30° tan 60° = cot 60° cot 30° = tan 45°. 

7. Show that tan 60° sin 2 45° = cos 30° sin 90°. 

8. Show that cos a tan a = sin a ; sin j3 cot f3 = cos j3. 

9. Show that - - tan230 ° = C os 60° = * cos 0°. 

1 + tan 2 30° * 

10. Show that (tan y + cot y) sin y cos y = 1. 

EXAMPLES. 

1. Show that sin 30° cos 60° + cos 30° sin 60° = sin 90°. 

2. Show that cos 60° cos 30° + sin 60° sin 30° = cos 30°. 

3. Show that sin 45° cos 0° - cos 45° sin 0° = cos 45°. 

4. Show that cos 2 45° - sin 2 45° = cos 90°. 

5. Show that tan 45° + tan 0° = ^ 

1 - tan 45° tan 0° 

If A = 60°, verify 



c . A 1 — cos A 

6. sin — = \ 

2 \ 2 



= Jl- L cos^. 
u + cos A 



7. tau — 
2 

8. cos A = 2 cos 2 ^.-1 = 1-2 sin 2 — • 

2 2 



If a = 0°, = 30°, y = 45°, 8 = 60°, e = 90°, find the values of 
9. sin j3 + cos 8. 

10. cos (3 + tan 8. 

11. sin j3 cos 8 + cos /? sin 8 — sin e. 

12. (sin (3 + sin f ) (cos cj + cos 8) — 4 sin a (cos y + sin c) . 



VARIATIONS IN THE FUNCTIONS. 



19 



24. Variations in the trigonometric functions. 

Signs. Thus far no account has been taken of the signs of 
the functions. By the definitions it appears that these de- 
pend on the signs of x, y, and r. Now r is always positive, 
and from the figures it is seen that x is positive in the first 








Y 


Sin. + 




Sin. + 




Cos. + 
Tan. + 




Csc. + 




Cot. + 
Sec. + 


v' 


(X-) 




Csc. + 


JL 


(£+) 


°(y 


~) 




Tan. + 




Cos. + 




Cot. + 


Y' 


Sec. + 



Fig. 11 



and fourth quadrants, and y is positive in the first and 
second. Hence 

For an angle in the first quadrant all functions are positive, 
since #, y, r are positive. 

In the second quadrant x alone is negative, so that those 
functions whose ratios involve x, viz. cosine, tangent, co- 
tangent, secant, are negative; the others, sine and cosecant, 
are positive. 

In the third quadrant x and y are both negative, so that 
those functions involving r, viz. sine, cosine, secant, cosecant, 
are negative; the others, tangent and cotangent, are positive. 

In the fourth quadrant y is negative, so that sine, tangent, 
cotangent, cosecant are negative, and cosine and secant, positive. 

Values. In the triangle of reference of any angle, the 
hypotenuse r is never less than x or y. Then if r be taken of 
any fixed length, as the angle varies, the base and perpen- 
dicular of the triangle of reference may each vary in length 

x u 

from to r. Hence the ratios - and - can never be greater 

r r & 

r r 



than 1, nor if x and y are negative, less than —1; and 



x y 



20 



PLANE TRIGONOMETRY. 



cannot have values between + 1 and — 1. But the ratios 

u x 

£ and - may vary without limit, i.e. from + oo to — oo. 

x y 

Therefore the possible values of the functions of an angle 

are : 

sine and cosine between + 1 and — 1, 

i.e. sine and cosine cannot be numerically greater than 1; 

tangent and cotangent between + oo and — oo, 

i.e. tangent and cotangent may have any real value ; 

secant and cosecant between + oo and ■+■ 1, and — 1 and — oo, 

i.e. secant and cosecant may have any real values, except 
values between + 1 and — 1. 

These limits are indicated in the following figures. The 
student should carefully verify. 



^) Sin 90° = 1 
Cos 90= ±0 
Tan 90 = ±oo 




^\Sin 0=±0 

X 180X- 





j 

Sin. + 1 


0° 

Y 

+ i 






Cos. - 


+ 






Tan. — oo 


+00 




.8 
t+0, 


-1, -0 


+ o, 


i 
+ 1, 


-0, 


-1, +0 


O -o, 


*■ i. 




Sin. - 1 


-l 






Cos. - 


+ 






Tan. +co 


-co 

Y' 





I 

-0X° C 



370° 



Fig. 18. 



25. In tracing the changes in the values of the functions as 

a changes from 0° to 360°, consider the revolving line r as 

of fixed length. Then x and y may have any length between 

and r. 

y = 

r r 



Sine. At 0°, sin a 



0. As a increases through 



y ... 



the first quadrant, y increases from to r, whence - increases 
from to 1. In passing to 180° sin a decreases from 1 to 0, 



VARIATIONS IN THE FUNCTIONS. 21 

since y decreases from r to 0. As a passes through 180°, y 
changes sign, and in the third quadrant decreases to nega- 
tive r, so that sin a decreases from to — 1. In the fourth 
quadrant y increases from negative r to 0, and hence sin a 
increases from — 1 to 0. 

Cosine depends on changing values of x. Show that, 
as a increases from 0° to 360°, cos a varies in the four 
quadrants as follows: 1 to 0, to — 1, — 1 to 0, to 1. 

Tangent depends on changing values of both y and x. 

At 0°, y = 0, x = r, at 180°, y = 0, x = - r, 

at 90°, x = 0, y =j r, at 270°, x = 0, y = - r. 

v 
Hence tan 0° = -^ = - = 0. As a passes to 90°, y increases 
x r 

to r, and x decreases to 0, so that tan a increases from to oo. 
As a passes through 90°, x changes sign, so that tan a 
changes from positive to negative by passing through oo. 
In the second quadrant x decreases to negative r, y to 0, and 
tan a passes from — oo to 0. As a passes through 180°, 
tan a changes from minus to plus by passing through 0, 
because at 180° y changes to minus. In the third quadrant 
tan« passes from to oo, changing sign at 270° by passing 
through oo, because at 270° x changes to plus. In the fourth 
quadrant tan a passes from — oo to 0. 

Cotangent. In like manner show that cot a passes through 
the values oo to 0, to — oo, oo to 0, to — oo, as a passes 
from 0° to 360°. 

Secant depends on x for its value. Noting the change 
in x as under cosine, we see that secant passes from 1 to oo, 

— oo to — 1, — 1 to — oo, oo to 1. 

Cosecant passes through the values oo to 1, 1 to oo, 

— oo to — 1, — 1 to — oo. 

The student should trace the changes in each function 
fully, as has been done for sine and tangent, giving the 
reasons at each step. 



22 



PLANE TRIGONOMETRY. 



a 


0° to 90° 


90° to 180° 


180° to 270° 


270° to 360° 


sin 


to 1 


1 to 


- to - 1 


- 1 to - 


cos 


1 to 


- to - 1 


- 1 to - 


to 1 


tan 


to oo 


— oo to — 


to oo 


— oo to — 


cot 


oo to 


— to — oo 


oo to 


— to — oo 


sec 


1 to 00 


— oo to — 1 


— 1 to — 00 


oo to 1 


esc 


oo to 1 


1 to 00 


— oo to — 1 


— 1 to — 00 



* 26. Graphic representation of functions. These variations 
are clearly brought out by graphic representations of the 
functions. Two cases will be considered : I, when a is a 
constant angle ; II, when a is a variable angle. 

I. When a is a constant angle. 

The trigonometric functions are ratios, pure numbers. 
By so choosing the triangle of reference that the denomi- 
nator of the ratio is a side of unit length, the side forming 
the numerator of that ratio will be a geometrical representa- 
tion of the value of that function, e.g. if in Fig. 19 r — 1, 



then sin 



r 1 



This may be done by making a a 



central angle in a circle of radius 1, and drawing triangles 
of reference as follows : 




Fig. 19. 



GRAPHIC REPRESENTATION OF FUNCTIONS. 23 



In all the figures 


AOP = 


- a, and 




sin a 


BP 
OP 


BP _ 

1 


BP, 




cos a 


OB 
OP 


OB 
1 


OB, 




tan a 


BP 

OB 


AD 

: OA 


.AD- 

1 


-AD, 



OA EC EC p/y 
COta = AB = OE = — = B °> 



sec a 


OP 
OB 


OD 
' OA 


OD _ 

1 


OD, 


esc a 


OP _ 
BP 


00 
' OE 


00 

1 


■00. 



It appears then that, by taking a radius 1, 

sine is represented by the perpendicular to the initial line, 
drawn from that line to the terminus of the arc sub- 
tending the given angle; 

cosine is represented by the line from the vertex of the 
angle to the foot of the sine ; 

tangent is represented by the geometrical tangent drawn 
from the origin of the arc to the terminal line, produced 
if necessary; 

cotangent is represented by the geometrical tangent drawn 
from a point 90° from the origin of the arc to the 
terminal line, produced if necessary ; 

secant is represented by the terminal line, or the terminal 
line produced, from the origin to its intersection with 
the tangent line ; 

cosecant is represented by the terminal line, or the terminal 
line produced, from the origin to its intersection with 
the cotangent line. 



24 



PLANE TRIGONOMETRY. 



These lines are not the functions, but in triangles drawn 
as explained their lengths are equal to the numerical values 
of the functions, and in this sense the lines may be said to 
represent the functions. It will be noticed also that their 
directions indicate the signs of the functions. Let the 
student by means of these representations verify the results 
of Arts. 24 and 25. 



II. When a is a variable angle. 

Take XX' and YY' as axes of reference, and let angle 
units be measured along the #-axis, and values of the func- 
tions parallel to the ?/-axis, as in Art. 3. We may write 
corresponding values of the angle and the functions thus : 

a=0 6 , 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 
sma = 0, J, JV2, iV3, 1, |V3, £V2, {, 0, - |, - |v% 

a= 240°, 270°, 300°, 315°, 330°, 360°, -30°, -45°, -60°, -90°, etc., 



»ina=-$V3, -1, -%VS, -|V2, -£, 0, 



iV2, -|V3, -1, etc. 



These values will be sufficient to determine the form of the 
curve representing the function. By taking angles between 

those above, and computing 
the values of the function, as 
given in mathematical tables, 
the form of the curve can be 
determined to any required 
degree of accuracy. Reduc- 
ing the above fractions to 
decimals, it will be convenient 
to make the ^/-units large in 
comparison with the #-units. 
In the figure one a>unit repre- 
sents 15°, and one y-unit 0.25. 
Measuring the angle values along the #-axis, and from these 
points of division measuring the corresponding values of sin a 
parallel to the ?/-axis, as in Art. 3, we have, approximately, 




Ciwves of Sine and Cosecant. 

Sine 

Cosecant - 

Fig. 20. 



GRAPHIC REPRESENTATION OF FUNCTIONS. 



25 



0X 1 = 30° = 2 units, OX 2 = 45° =3 units, 

X 1 F 1 = i =2 units, X 2 F 2 = 0.71 = 2.84 units, 

OX 3 = 60° =4 units, etc., 
X 3 Y 3 = 0.86 = 3.44 units, etc. 

We have now only to draw through the points Y v P" 2 , Y z . 
etc., thus determined, a continuous curve, and we have the 
sine-curve or sinusoid. 

The dotted curve in the figure is the cosecant curve. Let 
the student compute values, as above, and draw the curve. 

In like manner draw the cosine and secant curves, as 
follows : 



Curves of Cosine and Secant. 

Cosine 

Secant" - 

Fig. 21. 



Tangent curve. Compute values for the angle a and for 
tan «, as before : 

a = 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 
tan a = 0, iV3, 1, V3, ±», -V3, -1, -^V3, 0, iV3, 1, V3, ±», 



a = - 30°, - 45°, - 60°, - 90°, etc., 
tan#=— W3, —1, — a/3, ±20, etc. 



Then lay off the values of a and of tan a along the a?, and 
parallel to the ?/-axis, respectively. It will be noted that, 



26 



PLANE TRIGONOMETRY. 



as a approaches 90°, tan a increases to oo, and when a passes 
90°, tan a is negative. Hence the value is measured parallel 




Curves of Tangent and Cotangent. 

Tangent 

Cotangent — 

Fig. 22. 



to the ?/-axis downward, thus giving a discontinuous curve, 
as in the figure. 



* 27. The following principles are illustrated by the curves : 

1. The sine and cosine are continuous for varying values 
of the angle, and lie within the limits + 1 and — 1. Sine 
changes sign as the angle passes through 180°, 360°, •••, 
n 180°, while cosine changes sign as the angle passes through 
90°, 270°, •••, (2^ + 1) 90°. Tangent and cotangent are 
discontinuous, the one as the angle approaches 90°, 270°, •••, 
(2w + l) 90°, the other as the angle approaches 180°, 360°, •••, 
wl80 o , and each changes sign as the angle passes through 
these values. The limiting values of tangent and cotangent 
are + co and — oo. 

2. A line parallel to the ?/-axis cuts any of the curves in 
but one point, showing that for any value of a there is but 
one value of any function of a. But a line parallel to the 
#-axis cuts any of the curves in an indefinite number of 
points, if at all, showing that for any value of the function 
there are an indefinite number of values, if any, of a. 



GRAPHIC REPRESENTATION OF FUNCTIONS. 27 

3. The curves afford an excellent illustration of the varia- 
tions in sign and value of the functions, as a varies from 
to 360°, as discussed in Art. 25. Let the student trace these 
changes. 

4. From the curves it is evident that the functions are 
periodic, i.e. each increase of the angle through 360° in the 
case of the sine and cosine, or through 180° in the case of 
the tangent and cotangent, produces a portion of the curve 
like that produced by the first variation of the angle within 
those limits. 

5. The difference in rapidity of change of the functions 
at different values of a is important, and reference will be 
made to this in computations of triangles. (See Art. 64, 
Case III.) A glance at the curves shows that sine is chang- 
ing in value rapidly at 0°, 180°, etc., while near 90°, 270°, 
etc., the rate of change is slow. But cosine has a slow rate 
of change at 0°, 180°, etc., and a rapid rate at 90°, 270°, etc. 
Tangent and cotangent change rapidly throughout. 

Ex. Let the student discuss secant and cosecant curves. 

ORAL WORK. 

1. Express in radians 180°, 120°, 45°; in degrees, -^ radians, 2-n; 

f 7T, f 7T. 

2. If i of a right angle be the unit, what is the measure of | of a 
right angle? of 90°? of 135°? 

3. Which is greater, cos 30° or \ cos 60°? tan - or cot-? sin - or cos- ? 

5 2 6 3 4 4 

4. Express sin a in terms of sec a ; of tan a ; tan a in terms of cos a ; 
of sec a. 

5. Given sin a = f, find tan a. If tan a = 1, find sin a, esc a, cot a; 
also tan 2 a, sin 2 a, cos 2 a. 

6. If cos a = A, find sin -, tan — 

2 ' 2 2 

7. In what quadrant is angle t, if both sin t and cos t are minus ? if 
sin t is plus and cos t minus ? if tan t and cot t are both minus ? if sin t 
and esc t are of the same sign ? Why ? 

8. Of the numbers 3, f , — 5, — I, a, — b, oo, 0, which may be a value 
of sin jo? of sec j9? of tan pi Why? 



28 PLANE TRIGONOMETRY. 

EXAMPLES. 

1. If sin 26° 40' = 0.44880, find, correct to 0.00001, the cosine and 
tangent. 

2. If tan a = V3, and cot (3 = | V3, find sin a cos (3 — cos a sin /?. 

3 Evaluate sin 30 ° cot 30 ° - cos 6 0° tan 60° 
sin 90° cos 0° 

Prove the identities : 

4. tan ^1(1 -cot 2 ,4) + cot ,4(1 -tan 2 ^) = 0. 

5. (sin ,4 + sec^) 2 + (eosA + csc^4) 2 =(1 + sec A csc^4) 2 . 

6. sin 2 x cos x esc x — cos 3 x esc x sin 2 a; + cos 4 x sec x sin x = sin 3 # cos x 
+ cos 3 x sin x. 

7. tan 2 w + cot 2 w = sec 2 w esc 2 w — 2. 

8. sec 2 v + cos 2 v = 2 + tan 2 v sin 2 v. 

9. cos 2 * + 1 = 2 cos 3 i sec t + sin 2 *. 

10. csc 2 £ — sec 2 £ = cos 2 1 esc 2 1 — sin 2 1 sec 2 ?. 

11. The sine of an angle is ~ — ; find the other functions. 

m 2 + n 2 

12. If tan A + sin A = m, tan A — sin A — n, prove m 2 — n 2 = 4:Vmn. 

Solve for one function of the angle involved the equations : 

13. sin0 + 2cos0 = 1. 16. 2 sin 2 a: + cos a: - 1 = 0. 
cos a 3 17. sec 2 a; — 7 tan x — 9 = 0. 

' tana~2 18. 3 cscy + 10 cot y - 35 = 0. 

15. V3 esc 2 6 = 4 cot 0. 19. sin 2 y -|cosv- 1 = 0. 

20. a sec 2 to + b tan w + c — a = 0. 

21. If fELd: = V2, ^i =V3, find A and 5. 

sin i> tan B 

22. Find to five decimal places the arc which subtends the angle of 
1° at the centre of a circle whose radius is 4000 miles. 

23. If csc ,4 = |VB, find the other functions, when A lies between 

- and 7T. 
2 

24. In each of two triangles the angles are in G. P. The least angle 
of one of them is three times the least angle of the other, and the sum of 
the greatest angles is 240°. Find the circular measure of each of the 
angles. 



CHAPTER III. 

FUNCTIONS OF ANY ANGLE — IN VERSE FUNCTIONS. 

28. By an examination of the figure of Art. 24 it is seen 
that all the fundamental relations between the functions hold 
true for any value of a. The table of Art. 16 expresses the 
functions of a, whatever be its magnitude, in terms of each 
of the other functions of that angle if the ± sign be prefixed 
to the radicals. 

The definitions of the trigonometric functions (Art. 12) 
apply to angles of any size and sign, but it is always possible 
to express the functions of any angle in terms of the func- 
tions of a positive acute angle. 

The functions of any angle 0, greater than 360°, are the 
same as those of 6 ± n • 360°, since 6 and 6 ± n • 360° have 
the same triangle of reference. Thus the functions of 390°, 
or of 750°, are the same as the functions of 390° — 360°, or 
of 750°— 2-360°, i.e. of 30°, as is at once seen by drawing a 
figure. So also the functions of — 315°, or of — 675° are 
the same as those of - 315° + 360°, or of - 675° + 2-360°, 
i.e. of 45°. 

For functions of angles less than 360° the relations of this 
chapter are important. 

29. To find the relations of the functions of — 0, 90° ± 0, 
180° ± 0, and 270° ± to the functions of 0, 6 being any angle. 

Four sets of figures are drawn, I for 6 an acute angle, II 
for obtuse, III for an angle of the third quadrant, and 
IV for an angle of the fourth quadrant. 

In every case generate the angles forming the compound 
angles separately, i.e. turn the revolving line first through 

29 



30 PLANE TRIGONOMETRY. 

(a) (b) (c) 

180°± 




II 




in 





IV 




FUNCTIONS OF ANY ANGLE. 



31 







90 


ifl 






fx' 


Xf\ 






\ ^J 




1 




\t§y 




y 


Ui 


£ 


2/' 





/ 


\rt 


r/ 




2/ 




-Scv\ 


w 


a?'V 


Jx< 




*>o 


+B 





III 



/ 


\rv 




ry^ 


2/ 


•V(\ 




fxf' 




\^So 


+6 . 






90 


-Q 






Fig. 23. 



32 PLANE TRIGONOMETRY. 

0°, 90°, 180°, or 270°, and then from this position through 
0, or — 0, as the case may be. Form the triangles of refer- 
ence for (a) the angle 0, (6) - 0, (c) 180° ± 0, (d) 90° ± 0, 
00 270° ±0. 

The triangles of reference (a), (J), 00, 0-0 1 an ^ 00 > i* 1 
each of the four sets of figures, I, II, III, IV, are similar, 
being mutually equiangular, since all have a right angle and 
one acute angle equal each to each. Hence the sides a?, ?/, r 
of the triangles 00 are homologous to x\ y\ r r of the cor- 
responding triangles (b} and (c), but to y\ x\ r 1 ', of the 
corresponding triangles (d) and (e). For the sides x of 
triangle (a) and a;' of the triangles (5) and (c) are opposite 
equal angles, and hence are homologous, but the sides y' are 
opposite this same angle in triangles (cT) and (e), and there- 
fore sides y' of (eT) and O) are homologous to x of (a). 

Attending to the signs of x and x\ y and y f in the similar 
triangles (a) and (5), 

sin(-0)=^ = -|=-sin0, 

cos(-6>)=^ = ^ =cos6> ' 

tan ( - 0) = ^ = - ^ = - tan 0. 

a?' a; 

Also in the similar triangles (a) and (<?), 

2/ 



sin (180° -0) = ^ 
cos(18O°-0) = - 



tan(18O°-0)=^ 



= * = sin0, 
r 

= = — cos 0, 

r 

= - $- = - tan 6. 



In like manner show that 

sin(18O° + 0) = -sin0, 
cos (180° + 0) = - cos 0, 
tan (180° + 0)= tan 0. 



FUNCTIONS OF ANY ANGLE. 33 

Again, in the similar triangles (<%) and (cf), 



sin (90° + 0) = y -, = - = cos (9, 



r r 



y 



cos (90° + 0) =- = - £ = - sin 0, 

tan (90° + 0)= ^ = - -= - cot 0. 

x y 

Show that 

sin (90° - 0) = cos 0, 

cos (90° - 0) = sin 0, 

tan(9O°-0) = cot0. 
Finally, from the similar triangles (a) and (e), show that 

sin (27O°±0)=-cos0, 

cos(27O°±0)=±sin0, 

tan (270° ± 0) = T cot 0. 

From the reciprocal relations the student can at once 
write the corresponding relations for secant, cosecant, and 
cotangent. 

30. Since in each of the four cases x\ y' of triangles 
(5) and (V) are homologous to z, y of triangle (&), while 
z', y' of the triangles (cT) and (e) are homologous to y, x 
of triangle (<%), we may express the relations of the last 
article thus : 

The functions of \ -, ™ a correspond to the same functions 

of 0, while those of \ 9 -q , a correspond to the co-functions 
of 0, due attention being paid to the si 



The student can readily determine the sign in any given 
case, whether be acute or obtuse, by considering in what 
quadrant the compound angle, 90° ± 0, 180° ± 0, etc., would 



34 PLANE TRIGONOMETRY. 

lie if were an acute angle, and prefixing to the correspond- 
ing functions of 6 the signs of the respective functions for 
an angle in that quadrant. Thus 90° + 0, if 6 be acute, is 
an angle of the second quadrant, so that sine and cosecant 
are plus, the other functions minus. It will be seen that 
sin (90° + 0y= + cos <9, cos (90° + 0)= - sin<9, etc., and this 
will be true whatever be the magnitude of 0. It will assist 
in fixing in the memory these important relations to notice 
that when in the compound angle 6 is measured from the 
^-axis, as in 90° ± 0, 270° ± 0, the functions of one angle 
correspond to the co-functions of the other, but when in the 
compound angle is measured from the #-axis, as in ± 0, 
180° ± 0, then the functions of one angle correspond to the 
same functions of the other. 

These relations, as has been noted in Art. 28, can be 
extended to angles greater than 360°, and it may be stated 
generally that 

function = ± function (2 n • 90° ± 0), 

function = ± co-function [(2 n + 1) 90° ± 0]. 

Computation tables contain angles less than 90° only. The chief 
utility of the above relations will be the reduction of functions of angles 
greater than 90° to functions of acute angles. Thus, to find tan 130° 20', 
look in the tables for cot 40° 20', or for tan 49° 40'. Why? 

Ex. 1. What angles less than 360° have the same numerical cosine 
as 20° ? 

cos 20° = - cos (180° ± 20°) = cos (360° - 20°). 

.-. 200°, 160°, 340° have the same cosine numerically as 20°. 

2. Find the functions of 135°; of 210°. 

sin 135° = sin (90° + 45°) = cos 45° = %V2, 

cos 135° = cos (180° - 45°) = - cos 45° = - \ V2, etc. 

sin 210° = sin (180° + 30°) = - sin 30° = - \. 

Let the student give the other functions for each angle. 



INVERSE FUNCTIONS. 35 

ORAL WORK. 

1. Determine the sine and tangent of each of the following angles : 
30°, 120°, - 30°, - 60°, 1 tt, 2f tt, - 135°, - tt. 

2. Which is the greater, sin 30° or sin (- 30°) ? tan 135° or tan 45°? 
cos 60° or cos( - 60°) ? sin 22° 30' or cos 67° 30' ? 

3. What positive angle has the same tangent as — ? the same sine 
as 50°? 3 

4. If tan = - 1, find sin 0. 

5. Find sin 510°, cos(- 60°), tan 150°. 

6. Reduce in two ways to functions of a positive acute angle, cos 122° 
tan 140° 30', sin (-60°). 

7. Find all positive values of x, less than 360°, satisfying the fol- 
lowing equations : cos x = cos 45°, sin 2 x = sin 10°, tan 3 x = tan 60°, 
sin x = sin 30°, tan x = tan 135°. 

8. What angles are determined when (a) sine and cosine are + ? 
(b) cotangent and sine are — ? (c) sine + and cosine — ? (d) cosine — 
and cotangent + ? 

INVERSE FUNCTIONS. 

31. That a is the sine of an angle 6 may be expressed in 
two ways, viz., sin = a, or, inversely, 6 = sin -1 a, the latter 
being read, 6 equals an angle whose sine is a, or, more briefly, 
6 is the anti-sine of a. 

The notation sin -1 a, cos -1 a, tan -1 a, etc., is not a fortunate one, but 
is so generally accepted that a change is not probable. The symbol may 
have been suggested from the fact that if ax = b, then x = a' 1 b, whence, 
by analogy, if sin = a, = sin -1 a. But the likeness is an analogy only, 
for there is no similarity in meaning. Sin -1 a is an angle 0, where sin = a, 

and is entirely different from (sin a) -1 = — . In Europe the symbols 

arc sin a, arc cos a, etc., are employed. 

32. Principal value. We have found that in sin 6 = a, 
for any value of 0, a can have but one value ; but in 
6 = sin -1 a, for any value of a there are an indefinite number 
of values of d (Art. 27, 2). 

Thus, when sin 6 = a, if a = 1 6 may be 30°, 150°, 390°, 
510°, - 330°, etc., or, in general^ nir +(- l) w 30°. 

In the solution of problems involving inverse functions, 



36 



PLANE TRIGONOMETRY. 



the numerically least of these angles, called the principal 
value, is always used ; i.e. we understand that sin -1 a, tan -1 a, 
are angles between + 90° and — 90°, while the limits of 
cos" 1 a are 0° and 180°. 

Thus, sin" 1 f=30°, sin- 1 (-i)=-30°, cos" 1 \ = 60°, 
cos- 1 (-l)=120°. 



ORAL WORK. 

How many degrees in each of the following angles? How many 
radians ? 



- , V3 

1. cos x — 

o 


? 


7. 


tan-!V3? 


2, 

2. tan- 1 !? 


8. 


COS" 1 *)? 


3. cot -1 (-V3)? 


9. 


sin" 1 !? 


4. sin-X-iV^)? 


10. 


tan" 1 *)? 


5. cos-^-- 1 ^)? 






«^(-# 


11. 
12. 


tan-^-l)? 
sin-^-l)? 


Find the values of the functions : 




13. siri(tan- 1 $v'3). 


19. 


cos(sin -1 0). 


14. tan (cos -1 1). 


20. 


sin (cos -1 [— 1]). 


15. tan(cot _1 [— oo]). 


21. 


cos (cot -1 V3). 


16. cos(tan _1 Go). 






17. sin(sin- 1 i^2)- 


22. 


tan(sin" 1 [-l]). 


18. tan (tan -1 a;). 


23. 


sin (tan -1 [- 1]). 


y* 


Ex 


. 1. Construct cot 


-H, 


// 


Construct the right triangle xyr, so that x = 4, 


/ 


y = 3 > 


whence angle xr - 


= cot -1 f. 


JS y 


=3 








2. 


Find cos (tan -1 T 8 ^). 


/ 


Let 


6 = tan -1 T 8 ¥ , whe 
tan 6 = T 8 5, a 


nee 


X=i 




nd cos = |f . 


Fig. 24. 


••. cos 6 = cos 


(tan- 1 T ^)=if. 


3. If $ = esc -1 a, prove 


_, Va 2 - 1 

= cos 1 . 

a 




esc 6 


• a 1 

= a ; .-. sm v = - 








a 





and 



cos 



e = jnn = vsei, or 9 = cos .x vo^i. 



EXAMPLES. 37 



EXAMPLES. 

1. Construct sm _1 §, tan -1 ^, cos -1 (— £). 

2. Find tan(sin _1 T 5 g ), sin (tan -1 T 5 ^). 

3. If 6 = sin -1 a, prove = tan -1 



VI -a 2 

4. Show that sin -1 a = 90° — cos -1 a. 

5. Prove tan" 1 ^ + cot" 1 V3 = -• 

6. Prove tan- 1 ( sin - ] = cos- 1 A V2. 



7. What angles, less than 360°, have the same tangent numerically 
as 10°? 

8. Given tan 143° 22' = - 0.74357; find, correct to 0.00001, sine and 
cosine. 

9. If cot 2 (90° + /?) + esc (90° - /3) - 1 = 0, find tan (3. 

10. Find all positive values of x, less than 360°, when sin x = sin 22° 30' ; 
when tan 2 x = tan 60°. 

11. When is sin x = possible, and when impossible ? 

2 ab 

12. Verify sin" 1 \ + cos" 1 — + tan" 1 V3 = sin" 1 — • 

13. What values of x will satisfy sin" 1 (.T 2 — x) = 30° ? 

14. If tan 2 - sec 2 a = 1, prove sec 6 + tan 3 esc = (3 + tan 2 ay. 

15. Prove sin A (1 + tan ^4) + cos A (1 + cot ^1) = sec A + esc A. 

16. Solve the simultaneous equations : 

sin" 1 (2 x + 3 y) = 30° and 3 a; + 2 y = 2. 



17. Verify (a) tan 60° = J 1 = cos 12 Q 

J v J n + cos 12 



+ cos 120 c 
1 - tan 2 30° 
1 + tan 2 30°' 

(c) 2 sin 2 60° = 1 - cos 120 c 



(&) cos 60° 



18. Show that the cosine of the complement of - equals the sine of 

Q 
the supplement of — 



38 PLANE TRIGONOMETRY. 

REVIEW. 

Before leaving a problem the student should review and master all 
principles involved. 

1. Construct cos -1 T 8 7 ; sin _1 (— f ) ; tan -1 2. 

2. Find cos (sin -1 f ) ; tan (cos -1 [ — J] ) . 

3. Prove cot -1 a = cos -1 — = ^ == - 

4. Given a = cot" 1 f , find tan a + sin (90° + a). 

5. Find tan f sin -1 \ + cos -1 J • 

6. State the fundamental relations between the trigonometric func- 
tions in terms of the inverse functions. Thus, 



sin -1 a — esc -1 -, sin -1 a = cos _1 Vl — a 2 , etc. 

7. Find all the angles, less than 360°, whose cosine equals sin 120°. 

8. Given cot -1 2.8449, find the sine and cosine of the angle, correct 
to 0.0001. 

9. If tan 2 (180° -$)- sec (180° + 0) = 5, find cos 0. 

in t-p • a 2 -c a tan 2 + cos 2 

10. If sin 6 = 1 find ■ T — • 

tan 2 — cos 2 

11. Is sin x — 2 cos £ + 3 sin a: — 6 = 0a possible equation? 

12. Verify (a) sin 60° = 2 tan 30 ° . 

J v J 1 + tan 2 30° 

(b) 2 cos 2 60 D = 1 + cos 120°. 

(c) cos 60° - cos 90° = 2 cos 2 30° - 2 cos 2 45°. 

13. If sin x = ^ — ~i~ ~ ) — find sec x an( j tan x. 

a 2 + 2ab + 2 b 2 

14. .Prove 1 ± sin ° - cos ° + 1 + sinfl + cosfl = g cgc $ 

1 + sin + cos 6 1 + sin — cos 

15. Prove 

cos 45° + cos 135° + cos 30° + cos 150° - cos 210° + cos 270° = sin 60°. 

16. If tan = , prove that 

Va 2 - b 2 

sin 0(1 + tan 0) + cos 0(1 + cot 0) - sec = |- 

17. Solve sin 2 x + sin 2 (x + 90°) + sin 2 (x + 180°) = 1. 



EXAMPLES. 39 



18. Given cos 2 cc = m sin a — n, find sin «. 

19. If sin 2 £= — ?— , find/?. 

^ 2 sec j8' ^ 

20. Given tan 238° = 1.6, find sin 148°. 

21. Prove tan" 1 m + cot" 1 m = 90°. 

22. Find sin (sin -1 j9 + cos -1 /)) . 

23. Solve cot 2 (2 esc 6 - 3) + 3 (esc (9 - 1) = 0. 

24. Prove sin 2 a sec 2 /3 + tan 2 fi cos 2 a = sin 2 a + tan 2 /?. 

25. Prove cos 6 F + sin 6 V = 1 - 3 sin 2 F + 3 sin 4 F. 

26. What values of .4 satisfy sin 2 A = cos 3 A ? 



27. If tan C = — — , and tan D = A/- 921^L y fi n d tan D in terms 

aim. m n + cosC 

28. If sin x — cos x + 4 cos 2 £ = 2, find tan x ; sec £. 

1 — 2 cos 2 £ 

29. Does the value of sec x, derived from sec 2 x = , give a 

possible value of x ? "~ cos x 

30. Prove 

[cot (90° - A)- tan (90° + A)] [sin (180° - A) sin (90° + A)~] = 1. 

31. Prove (1 + sin^4) 2 [cotyl +2 sec ^1(1 -esc A)] + esc A cos 3 A =0. 

32. Given sin x = m sin ?/, and tan x = n tan ?/, find cos x and cos y. 

33. Given cot 201° = 2.6, find cos 111°. 

34. Find the value of 

cos- 1 \ + sin" 1 iV2 + csc- 1 ( - 1) + tan" 1 1-2 cot" 1 ^. 

35. Solve 2 cos 2 6 + 11 sin 6 - 7 = 0. 

36. Prove 

cos 2 B + cos 2 {B + 90°) + cos 2 (B + 180°) + cos 2 (B + 270°) = 2. 



CHAPTER IV. 

COMPUTATION TABLES. 

33. Natural functions. It has been noted that the trigo- 
nometric functions of angles are numbers, but the values 
were found for only a few angles, viz. 0°, 30°, 45°, 60°, 
90°, etc. In computations, however, it is necessary to know 
the values of the functions of any angle, and tables have 
been prepared giving the numerical values of the functions 
of all angles between 0° and 90° to every minute. In 
these tables the functions of any given angle, and con- 
versely the angle corresponding to any given function, can 
be found to any required degree of accuracy ; e.g. by look- 
ing in the tables we find sin 24° 26' = 0.41363, and also 
1.6415 = tan 58° 39'. These numbers are called the natural 
functions, as distinguished from their logarithms, which are 
called the logarithmic functions of the angles. 

Ex. 1. Find from the tables of natural functions : 

sin 35° 14'; cos 54° 46'; tan 78° 29' ; cos 112° 58'; sin 135°. 

2. Find the angles less than 180° corresponding to : 
sin" 1 0.37865; cos" 1 0.37865; tan"* 0.58670; cos" 1 0.00291 ; sin" 1 0.99999. 

34. Logarithms. The arithmetical processes of multi- 
plication, division, involution, and evolution, are greatly 
abridged by the use of tables of logarithms of numbers 
and of the trigonometric ratios, which are numbers. The 
principles involved are illustrated in the following table : 

Write in parallel columns a geometrical progression having 
the ratio 2, and an arithmetical progression having the dif- 
ference 1, as follows : 

40 



LOGARITHMS. 



41 



G. P. 


A. P. 


1 





2 


1 


4 


2 


8 


3 


16 


4 


32 


5 


64 


6 


128 


7 


256 


8 


512 


9 


1024 


10 


2048 


11 


4096 


12 


8192 


13 


16384 


14 


32768 


15 


655S6 


16 


131072 


17 


262144 


18 


524288 


19 


1048576 


20 



It will be perceived that the numbers in 
the second column are the indices of the 
powers of 2 producing the corresponding 
numbers in the first column, thus : 2 6 = 64, 
2 n = 2048, 2 18 = 262144, etc. The use of 
such a table will be illustrated by examples. 

Ex. 1. Multiply 8192 by 128. 

From the table, 8192 = 2 13 , 128 = W. Then by 
actual multiplication, 8192 x 128 = 1048576, or by the 
law of indices, 2 13 x 2? = 2 20 = 1048576 (from table). 

Notice that the simple operation of addition is sub- 
stituted for multiplication by adding the numbers in 
the second column opposite the given factors in the 
first column. This sum corresponds to the number 
in the first column which is the required product. 

2. Divide 16384 by 512. 

16384 -f- 512 = 32, which corresponds to the result 
obtained by use of the table, or 2 14 -~ 2 9 = 2 5 = 32. 
The operation of subtraction takes the place of 
division. 



3. Find V262144. 



V262144 = V2 18 



= 23 = 



In the table, 262144 is opposite 18. 18 - 6 = 3, 
which is opposite 8, the required root ; i.e. simple division takes the 
place of the tedious process of evolution. 

4. Cube 64. 6. Find ^32768. 



5. Multiply 256 by 4096. 



7. Divide 1048576 by 32768. 



35. The above table can be made, as complete as desired 
by continually inserting between successive numbers in the 
first column the geometrical mean, and between the opposite 
numbers in the second, the arithmetical mean, but in prac- 
tice logarithms are computed by other methods. The num- 
bers in the second column are called the logarithms of the 
numbers opposite in the first column. 2 is called the base of 
this system, so that the logarithm of a number is the exponent 
by which the base is affected to produce the number. 



42 PLANE TRIGONOMETRY. 

Thus, the logarithm of 512 to the base 2 is 9, since 
2 9 = 512. 

Logarithms were invented by a Scotchman, John Napier, early in the 
seventeenth century, but his method of constructing tables was different 
from the above. See Encyc. Brit., art. "Logarithms" for an exceedingly 
interesting account. De Morgan says that by the aid of logarithms the 
labor of computing has been reduced for the mathematician to about 
one-tenth part of the previous expense of time and labor, while Laplace 
has said that John Napier, by the invention of logarithms, lengthened 
the life of the astronomer by one-half. 

Columns similar to those above might be formed with any- 
other number as base. For practical purposes, however, 10 
is always taken as the base of the system, called the common 
system, in distinction from the natural system, of which the 
base is 2.71828 •••, the value of the exponential series (Higher 
Algebra). The natural system is used in theoretical discus- 
sions. It follows that common logarithms are indices, positive 
or negative, of the powers of 10. 

Thus, 10 3 = 1000 ; i.e. log 1000 = 3 ; 

10- 2 = ^=0.01; i.e. log 0.01 = -2. 

36. Characteristic and mantissa. Clearly most numbers 
are not integral powers of 10. Thus 300 is more than the 
second and less than the third power of 10, so that 

log 300 = 2 plus a decimal. 

Evidently the logarithms of numbers generally consist of 
an integral and a decimal part, called respectively the charac- 
teristic and the mantissa of the logarithms. 

37. Characteristic law. The characteristic of the loga- 
rithm of a number is independent of the digits composing 
the number, but depends on the position of the decimal 
point, and is found by counting the number of places the first 
significant figure in the number is removed from the units' 1 
place, being positive or negative according as the first significant 



LOGARITHMS. 43 

figure is at the left or the right of units' place. This follows 
from the fact that common logarithms are indices of powers 
of 10, and that 10% n being a positive integer, contains n -\-l 
places, while 10 _/l contains n — 1 zeros at the right of units' 
place. Thus in 146.043 the first significant figure is two 
places at the left of units' place ; the characteristic of log 
146.043 is therefore 2. In 0.00379 the first significant digit 
is three places at the right of units' place, and the charac- 
teristic of log 0.00379 is - 3. 

To avoid the use of negative characteristics, such charac- 
teristics are increased by 10, and — 10 is written after the 
logarithm. Thus, instead of log 0.00811 = 3.90902, write 
7.90902 — 10. In practice the — 10 is generally not written, 
but it must always be remembered and accounted for in the 
result. 

Ex. Determine the characteristic of the logarithm of : 
1 ; 46 ; 0.009 ; 14796.4 ; 230.001 ; 10 5 x 76 ; 0.525 ; 1.03 ; 0.000426. 

38. Mantissa law. The mantissa of the logarithm of a 
number is independent of the position of the decimal point, 
but depends on the digits composing the number, is always 
positive, and is found in the tables. 

For, moving the decimal point multiplies or divides a 
number by an integral power of 10, i.e. adds to or subtracts 
from the logarithm an integer, and hence does not affect the 
mantissa. Thus, 

log 225.67 = log 225.67, 

log 2256.7 = log 225.67 xlO 1 = log 225.67 + 1, 
log 22567.0 = log 225.67 x 10 2 = log 225.67 + 2, 
log 22.567 = log 225.67 x 10" 1 = log 225.67 +(-1), 
log 0.22567 = log 225.67 x 10" 3 = log 225.67 +(- 3), 

so that the mantissse of the logarithms of all numbers com- 
posed of the digits 22567 in that order are the same, .35347. 
Moving the decimal point affects the characteristic only. 
The student must remember that the mantissa is always positive. 



44 PLANE TRIGONOMETRY. 

Log 0.0022567 is never written - 3 +.35347, but 3.35347, the minus 
sign being written above to indicate that the characteristic alone is nega- 
tive. In computations negative characteristics are avoided by adding 
and subtracting 10, as has been explained. 

39. We may now define the logarithm of a number as the 
index of the power to which a fixed number, called the base, 
must be raised to produce the given number. 

Thus, a x = b, and x = log a b (where log a 5 is read logarithm 
of b to the base a) are equivalent expressions. The relation 
between base, logarithm, and number is always 

(base) log = number. 

To illustrate : log 2 8 = 3 is the same as 2 3 = 8; log 3 81 = 4and 
3 4 = 81 are equivalent expressions ; and so are log 10 1000 = 3 
and 10 3 = 1000, and log 10 0.001= -3 and 10" 3 = 0.001. 

Find the value of : 

log 4 64; log 6 125; log 3 243 ; log^O) 1 ; log 27 3 ; log.l. 

40. From the definition it follows that the laws of indices 
apply to logarithms, and we have : 

I. The logarithm of a product equals the sum of the loga- 
rithms of the factors. 

II. The logarithm of a quotient equals the logarithm of the 
dividend minus the logarithm of the divisor. 

III. The logarithm of a power equals the index of the 
power times the logarithm of the number. 

IV. The logarithm of a root equals the logarithm of the 
number divided by the index of the root. 

For if a x — n and a y = m, 

then n x m = a x+p , .'. log nm = x + y = log n + logm; 

n 
and n-i-m = a x ~ y , .'.log— = x — y = \ogn — logm; 

also n r = (a x ~) r = a rx , .'. log n r =rx = r x log n ; 

- 1 

finally, \Jn = -Va x = a r , .*. log -y/n = - = - log n. 



LOGARITHMS. 45 

EXAMPLES. 
Given log 2 = 0.30103, log 3 = 0.47712, log 5 = 0.69897, find : 

1. log 4. 4. log 9. 7. log 153. 1Q log V||. 

2. log 6. 5. log 25. 8. logf. 

3. log 10. 6. logV3. 9- logl5x9. 



•• lo gN /: 



.. log % /*~ A 58 . 
2 4 xl0 



USE OF TABLES. 
41. To find the logarithm of a number. 

First. Find the characteristic, as in Art. 37. 

Second. Find the mantissa in the tables, thus : 

(a) When the number consists of not more than four 
figures. 

In the column iV of the tables find the first three figures, 
and in the row N the fourth figure of the number. The 
mantissa of the logarithm will be found in the row opposite 
the first three figures and in the column of the fourth figure. 

Illustration. Find log 42. 38. 

The characteristic is 1. (Why ?) 

In the table in column N find the figures 423, and on the 
same page in row N the figure 8. The last three figures of 
the mantissa, 716, lie at the intersection of column 8 and 
row 423. To make the tables more compact the first two 
figures of the mantissa, 62, are printed in column only. 
Then log 42.38 = 1.62716. 

Find log 0.8734 = 1.94121, 

log 3.5 = log 3.500 = 0.54407, 
log 36350 =4.56050. 

(5) When the number consists of more than four figures. 

Find the mantissa of the logarithm of the number com- 
posed of the first four figures as above. To correct for the 
remaining figures we interpolate by means of the principle of 
proportional parts, according to which it is assumed that, for 
differences small as compared with the numbers, the differences 



46 PLANE TRIGONOMETRY. 

between several numbers are proportional to the differences be- 
tween their logarithms. 

The theorem is only approximately correct, but its use 
leads to results accurate enough for ordinary computations. 

Ex. 1. To find log 89.4562. 

As above, mantissa of log 894500 = 0.95158, 

mantissa of log 894600 = 0.95163, 

.-. log 894600 - log 894500 = 0.00005, called the tabular difference. 

Let log 894562 - log 894500 = x hundred-thousandths. 

Now, by the principle of proportional parts, 

log 894562 - log 894500 = 894562 - 894500 
log 894600 - log 894500 894600 - 894500' 

or - = — , whence x = .62 of 5 = 3.1 
5 100 

.-. log 89.4562 = 1.95158 + 0.00003 = 1.95161, 

all figures after the fifth place being rejected in five-place tables. If, 
however, the sixth place be 5 or more, it is the practice to add 1 to the 
figure in the fifth place. Thus, if x = 0.0000456, we should call it 
0.00005, and add 5 to the mantissa. 

2. Find log 537.0643. 

To interpolate we have x : 9 = 643 : 1000, i.e. x = 5.787; 
.-. log 537.0643 = 2.72997 + 0.00006. 

3. Find log 0.0168342 = 2.22619. 

4. Find log 39642.7 = 4.59816. 

42. To find the number corresponding to a given logarithm. 

The characteristic of the logarithm determines the posi- 
tion of the decimal point (Art. 37). 

(a) If the mantissa is in the tables, the required number 
is found at once. 

Ex. 1. Find log -1 1.94621 (read, the number whose logarithm is 
1.94621). 

The mantissa is found in the tables at the intersection of row 883 and 
column 5. 

.-. log- 1 1.94621 = 88.35, 

the characteristic 1 showing that there are two integral places. 



LOGARITHMS. 47 

(5) If the exact mantissa of the given logarithm is not in 
the tables, the first four figures of the corresponding num- 
ber are found, and to these are annexed figures found by 
interpolating by means of the principle of proportional 
parts, as follows : 

Find the two successive mantissas between which the given 
mantissa lies. Then, by the principle of proportional parts, 
the amount to be added to the four figures already found is 
such a part of 1 as the difference between the successive 
mantissas is of the difference between the smaller of them 
and the given mantissa. 

2. Find log- 1 1.43764. 

Mantissa of log 2740 = 0.43775 

of log 2739 = 0.43759 
Differences 1 16 

Mantissa of log required number = 0.43764 

of log 2739 = 0.43759 

Differences x 5 

By p. p. x : 1 = 5 : 16 and x = T % = 0.3125. 

Annexing these figures, log" 1 1.43764 = 27.3931+. 

3. Find log" 1 1.48762. 

The differences in logarithms are 14, 6. 

... x = — = .428+, 
14 

and log- 1 1.48762 = 0.307343+. 

4. Find log 891.59; log 0.023 ; logi; log 0.1867; log V2. 

5. Find log" 1 2.21042 ; log" 1 0.55115 ; log- 1 1.89003. 

43. Logarithms of trigonometric functions. These might 
be found by first taking from the tables the natural func- 
tions of the given angle, and then the logarithms of these 
numbers. It is more expeditious, however, to use tables 
showing directly the logarithms of the functions of angles 
less than 90° to every minute. Functions of angles greater 
than 90° are reduced to functions of angles less than 90° by 



48 PLANE TRIGONOMETRY. 

the formulae of Art. 29. To make the work correct for 
seconds, or any fractional part of a minute, interpolation 
is necessary by the principle of proportional parts, thus : 

Ex. 1. Find log sin 28° 32' 21". 

In the table of logarithms of trigonometric functions, find 28° at the 
top of the page, and in the minute column at the left find 32'. Then 
under log sin column find log sin 28° 32' = 9.67913 - 10 

log sin 28° 33 = 9.67936 - 10 

Differences 1' 23 

By p. p. x : 23 = 21" : 60", i.e. x=—x23 = 8.05. 

j t f '60 

.-. log sin 28° 32' 21" = 9.67913 + 0.00008 - 10 
= 9.67921 - 10. 

Whenever functions of angles are less than unity, i.e. are decimals 
(as sine and cosine always are, except when equal to unity, and as tan- 
gent is for angles less than 45°), the characteristic of the logarithm will 
be negative, and, accordingly, 10 is always added in the tables, and it 
must be remembered that 10 is to be subtracted. Thus, in the example 
above, the characteristic of the logarithm is not 9, but 1, and the log- 
arithm is not 9.67913, as written in the tables, but 9.67913 — 10. 

2. Find log cos 67° 27' 50". 

In the table of logarithms at the foot of the page, find 67°, and in the 
minute column at the right, 27'. Then computing the difference as 
above, x = 25. 

But it must be noted that cosine decreases as the angle increases 
toward 90°. Hence, log cos 67° 27' 50" is less than log cos 67° 27', i.e. 
the difference 25 must be subtracted, so that 

log cos 67° 27' 50" = 9.58375 - 0.00025 - 10 
= 9.58350 - 10. 

44. To find the angle when the logarithm is given, find the 
successive logarithms between which the given logarithm 
lies, compute by the principle of proportional parts the 
seconds, and add them to the less of the two angles corre- 
sponding to the successive logarithms. This will not neces- 
sarily be the angle corresponding to the less of the two 
logarithms ; for, as has been seen, the number, and, therefore, 
the logarithm, may decrease as the angle increases. 






LOGARITHMS. 49 

Ex. 1. Find the angle whose log tan is 9.88091. 

log tan 37° 14' = 9.88079 - 10 
log tan 37° 15' = 9.88105 - 10 

Differences 60" 26 

log tan 37° 14' = 9.88079 - 10 
log tan angle required = 9.88091 — 10 

Differences x" 12 

.-. x : 60 = 12 : 26, or x" = i| x 60" = 28", approximately, and the 
angle is 37° 14' 28". 

2. Find the angle whose log cos = 9.82348. 

We find x = T % x 60" = 26", and the angle is 48° 14' 26". 

3. Show that log cos 25° 31' 20" = 9.95541 ; 

log sin 110° 25' 20" = 9.97181 ; 
log tan 49° 52' 10" = 0.07417. 

4. Show that the angle whose log tan is 9.92501 is 40° 4' 39"; whose 
log sin is 9.88365 is 49° 54' 18" ; whose log cos is 9.50828 is 71° 11' 49". 

45. Cologarithms. In examples involving multiplications 
and divisions it is more convenient, if n is any divisor, to 

add log - than to subtract log n. The logarithm of - is 
called the cologarithm of n. ' Since 

log — = log 1 — log n = — log n, 

it follows that colog n = — log n, i.e. logn subtracted from 
zero. To avoid negative results, add and subtract 10. 

Ex.1. Find colog 2963. 

log 1 = 10.00000 - 10 
log 2963= 3.47173 

.-. colog 2963 = 6.52827 - 10 

2. Find colog tan 16° 17'. 

log 1 = 10.00000 - 10 
log tan 16° 17' = 9.46554 - 10 

.-. colog tan 16° 17' = 0.53446 




50 PLANE TRIGONOMETRY. 

By means of the definitions of the trigonometric functions, the parts 
of a right triangle may be computed if any two parts, one of them being 
a side, are given. Thus, 

given a and A in the rt. triangle ABC. 

Then c = a -f- sin A, b = a -f- tan A, 
and 5 = 90°-^. 

Again, if a and b are given, then 

tan A = -, c = a + sin A, and B = 90° - A* 

b 

3. Given c = 25.643, B = 37° 25' 20", compute the other parts. 
A = 90° - 37° 25' 20" = 52° 34' 40". 

a = c cos B. b = a tan J5. 

log c = 1.40897 log a = 1.30889 

log cos B = 9.89992 log tan B = 9.88376 

log a = 1.30889 log b = 1.19265 

.-. a = 20.365. .-. b = 15.583. 

C%ec£: c* = a* + b* = 20.365 2 + 15.583 2 = 657.57 = 25.643 2 . 

4. Given b = 0.356, B = 63° 28' 40", compute the other parts. 

A = 26° 31' 20". 

b 

a = 

taiijB 

log b = 9.55145 

cologtan£ = 9.69816 

log c = 9.59974 log a = 9.24961 

c = 0.3979 a = 0.1777 

Check: c 2 - a 2 = 0.1583 - 0.03157 = 0.12673 = 6 2 . 

EXAMPLES. 

Compute the other parts : 

1. Given a = 9.325, A = 43° 22' 35". 

2. Given c = 240.32, a = 174.6. 

3. Given £ = 76° 14' 23", a = 147.53. 

4. Given a = 2789.42, b = 4632.19. 

5. Given c = 0.0213, A = 23° 14". 
Q. Given b = 2, c = 3. 





c = 


sin B 




log 6 = 


= 9.55145 


colog 


sinJ3 = 


= 0.04829 



CHAPTER V. 



APPLICATIONS. 



46. Many problems in measurements of heights and dis- 
tances may be solved by applying the preceding principles. 
By means of instruments certain distances and angles may 
be measured, and from the data thus determined other 
distances and angles computed. The most common instru-- 
ments are the chain, the transit, and the compass. 

The chain is used to measure distances. Two kinds are in 
use, the engineer's chain and the Ghunter's chain. They each 
contain 100 links, each link in the engineer's chain being 
12 inches long, and in the Gunter's 7.92 inches. 




Fig. 26. 

The transit is the instrument most used to measure hori- 
zontal angles, and with certain attachments to measure verti- 
cal angles. The figure shows the form of the instrument. 

51 



52 



PLANE TRIGONOMETRY. 



The mariner's compass is used to determine the directions, 
or bearings, of objects at sea. Each quadrant is divided 
into 8 parts, making the 32 points of the compass, so that 
each point contains 11° 15'. 






^ ^ 



^ o 







WW® 




s£££ 



£V:^ 








|^8 



3? 






g * 4 



Fig. 27. 



tf> 



47. The angle between the horizontal plane and the line 
of vision from the eye to the object is called the angle of 

elevation, or of depression, according 
as the object is above or below the 
observer. 

It is evident that the elevation 
angle of B, as seen from A, is equal 
to the depression angle of A, as seen from B, so that in the 
solution of examples the two angles are interchangeable. 




Fig. 28. 



PROBLEMS. 

48. Some of the more common problems met with in 
practice are illustrated by the following : 

To find the height of an object 
when the foot is accessible. 

The distance BO, and the eleva- 
tion angle B are measured, and x 
determined from the relation jf 



is 



x =BO tan B. 




Fig. 29. 



APPLICATIONS. 



53 



Ex. 1. The elevation angle of a cliff measured from a point 300 ft. 
from its base is found to be 30°. How high is the cliff? 



Then 



BC = 300, B = 30°. 
x = 300 • tan 30° = 



V3 = 100V3. 



2. From a point 175 ft. from the foot of a tree the elevation of the 
top is found to be 27° 19'. Find the height of the tree. 

The problem may be solved by the use of natural functions, or of 
logarithms. The work should be arranged for the solution before the 
tables are opened. Let the student complete. 



BC = 175. 
Then x = BC tan B. 

logBC = 
log tan B = 
log a: = 
.-. x = 90.39. 



B = 27° 19'. 

Or by natural functions, 
BC = 175 
tan B = 0.5165 



.-. x = 90.3875. 




To find the height of an object 
when the foot is inaccessible. 

Measure BB', and 6'. 

Then x= BO_ B£' + B'0 

cot 6 cot 6 

But B'C = x cot 6', whence substituting, 

BB' 

cot 6 - cot 0'' 

which is best solved by the use of the natural functions of 
and 0'. 

3. Measured from a certain point at its base the elevation of the 
peak of a mountain is 60°. At a distance of one mile directly from this 
point the elevation is 30°. Find the height of the mountain. 

BB' = 5280 ft., = 30°, & = 60°. 



5280 



cot 30° 
5280 



cot 30° - cot 60 



But y = x cot 60°. 
4572.48 ft. 



54 



PLANE TRIGONOMETRY. 



In surveying it is often necessary to make measurements 
across a stream or other obstacle too wide to be spanned by 

a single chain. 

To find the distance from C to a 
point B on the opposite side of a 
stream. 

At O measure a right angle, and 
take CA a convenient distance. 
Measure angle A, then 

BC=CA -tan. A. 
250 ft. 




Fig. 31. 

4. Find CB when angle A = 47° 16', and CA 

5. From a point due south of a kite its elevation is found to be 



42° 30' ; from a point 20 yds. due west 
from this point the elevation is 36° 24'. 
How high is the kite above the ground ? 

AB = x . cot 42° 30', 
A C = x . cot 36° 24', 
AC 2 - AB* = BC 2 = 400. 
.-. x 2 (cot 2 36° 24' - cot 2 42° 30') = 400, 
whence 

400_ and x _ J0_ = 24.84 yds. 



.6489 



.805 




jr 



\ 



E 



Fig. 32. 



EXAMPLES. 

1. What is the altitude of the sun when a tree 71.5 ft. high casts 

a shadow 37.75 ft. long? 

2. What is the height of a balloon directly over Ann Arbor when 
its elevation at Ypsilanti, 8 miles away, is 10° 15'? 

3. The Washington monument is 555 ft. high. How far apart are 
two observers who, from points due east, see the top of the monument 
at elevations of 23° 20' and 47° 30', respectively? 

4. A mountain peak is observed from the base and top of a tower 
200 ft. high. The elevation angles being 25° 30' and 23° 15', respec- 
tively, compute the height of the mountain above the base of the tower. 

5. From a point in the street between two buildings the elevation 
angles of the tops of the buildings are 30° and 60°. On moving across 



APPLICATIONS. 55 

the street 20 ft. toward the first building the elevation angles are found 
to be each 45°. Find the width of the street and the height of each 
building. 

6. From the peak of a mountain two towns are observed due south. 
The first is seen at a depression of 48° 40', and the second, 8 miles farther 
away and in the same horizontal plane, at a depression of 20° 50'. What 
is the height of the mountain above the plane ? 

7. A building 145 ft. long is observed from a point directly in front 
of one corner. The length of the building subtends tan -1 3, and the 
height tan -1 2. Find the height. 

8. An inaccessible object is observed to lie due N.E. After the ob- 
server has moved S.E. 2 miles, the object lies N.N.E. Find the distance 
of the object from each point of observation. 

9. Assuming the earth to be a sphere with a radius of 3963 miles, 
find the height of a lighthouse just visible from a point 15 miles distant 
at sea. 

10. The angle of elevation of a tower 120 ft. high due north of an 
observer was 35° ; what will be its angle of elevation from a point due 
west from the first point of observation 250 ft.? Also the distance of 
the observer from the base of the tower in each position ? 

11. A railway 5 miles long has a uniform grade of 2° 30' ; find the rise 
per mile. What is the grade when the road rises 70 ft. in one mile ? 

(The grade depends on the tangent of the angle.) 

12. The foot of a ladder is in the street at a point 30 ft. from the 
line of a building, and 'just reaches a window 22| ft. above the ground. 
By turning the ladder over it just reaches a window 36 ft. above the 
ground on the other side of the street. Find the breadth of the street. 

13. From a point 200 ft. from the base of the Forefathers' monument 
at Plymouth, the base and summit of the statue of Faith are at an eleva- 
tion of 12° 40' 48" and 22° 2' 53", respectively ; find the height of the 
statue and of the pedestal on which it stands. 

14. At a distance of 100 ft. measured in a horizontal plane from the 
foot of a tower, a flagstaff standing on the top of the tower subtends an 
angle of 8°, while the tower subtends an angle of 42° 20'. Find the 
length of the flagstaff. 

15. The length of a string attached to a kite is 300 ft. The kite's 
elevation is 56° 6'. Find the height of the kite. 

16. From two rocks at sea level, 50 ft. apart, the top of a cliff is ob- 
served in the same vertical plane with the rocks. The angles of eleva- 
tion of the cliff from the two rocks are 24° 40' and 32° 30'. What is the 
height of the cliff above the sea ? 



CHAPTER VI. 



GENERAL FORMULiE — TRIGONOMETRIC EQUATIONS 
AND IDENTITIES. 

49. Thus far functions of single angles only have been 
considered. Relations will now be developed to express 
functions of angles which are sums, differences, multiples, 
or sub-multiples of single angles in terms of the functions 
of the single angles from which they are formed. 

First it will be shown that, 

sin (a ± p) = sin a cos p ± cos a sin p, 
cos (a ± p) = cos a cos p T sin a sin p 

tan a ± tan p 

1 T tan a tan p 

The following cases must be considered : 

1. a, /3, a + /3 acute angles. 

2. a, /3, acute, but a -{- /3 an obtuse angle. 

3. Either a, or ft, or both, of any magnitude, positive or 
negative. 

The figures apply to cases 1 and 2. 



tan (a ± p) 





Vy 


C 








N 


\&f 


n 






C 


V 


A^- 










/s^a. 











D M 



Fig. 33 




Let the terminal line revolve through the angle a, and 
then through the angle /3, to the position OB, so that angle 



56 



GENERAL FORMULAE. 57 

XOB = a 4- /3. Through any point P in OB draw perpen- 
diculars to the sides of a, DP and (7P, and through draw 
a perpendicular and a parallel to OX, M and iV(7. 

Then the angle QUA = a (why?), and CNP is the triangle 
of reference for angle QCP = 90° + a. 

CNP is sometimes treated as the triangle of reference for angle CPN. 
The fallacy of this appears when we develop cos (a + /?), in which PC 
would be treated as both plus and minus. 

at • r , a\ ■ mil DP MO . NP 

Now sin(a + £)=flinX0£ = — = _ + _, 

or expressing in trigonometric ratios, 

= MO 00 NP OP 
00' OP OP ' OP 

= sin a cos ft + sin (90° + a) sin /3. 

Hence, since sin (90° + a) = cos a, we have 

sin (« + /3) = sin a cos /3 + cos a sin /3. 
In like manner 

cos (<* + £) = cos X0£ = — = _ + — , 

or expressing in trigonometric ratios, 

= QJf 00 ON OP 

00 ' OP op' op 

= cos a cos /3 + cos (90° + a) sin/3. 

And since cos (90° + a) = — sin ct, we have 

cos (a + /3) = cos a cos /3 — sin a sin ft. 

It will be noted that the wording of the demonstration ap- 
plies to both figures, the only difference being that when a-f j3 
is obtuse OD is negative. ON is negative in each figure. 

50. In the case, when a, or /3, or both, are of any magni- 
tude, positive or negative, figures may be constructed as 
before described by draiving through any point in the terminal 
line of ft a perpendicular to each side of a, and through the foot 
of the perpendicidar on the terminal line of a a perpendicular 
and a parallel to the initial line of a. Noting negative lines, 



58 PLANE TRIGONOMETRY. 

the demonstrations already given will be found to apply for 
all values of a and ft 

To make the proof complete by this method would require an unlim- 
ited number of figures, e.g. we might take a obtuse, both a and (5 obtuse, 
either or both greater than 180°, or than 360°, or negative angles, etc. 

Instead of this, however, the generality of the proposition 
is more readily shown algebraically, as follows : 

Let a! = 90° + a be any obtuse angle, and «, ft acute 
angles. 

Then 

sin (a' + 0) = sin (90° + a + ft) = cos (a + ft) 
= cos a cos ft — sin a sin ft 

= sin (90° + a) cos ft + cos (90° + a) sin /3(why?) 
= sin a' cos + cos a! sin ft 

In like manner, considering any obtuse angle 0' = 90° -f ft 
it can be shown that 

sin (V + 0') = sin a' cos ft + cos a' sin ft. 
Show that cos (V + ft) = cos a' cos ft — sin a' sin ft. 

By further substitutions, e.g. a" = 90° ± «', ft' = 90° ± ft, 
etc., it is clear that the above relations hold for all values, 
positive or negative, of the angles a and 0. 

Since a and may have any values, we may put — for ft 
and sin (a -f [— ft]) 

= sin (a — ft) = sin a cos (— ft) + cos a sin ( — 0) 

= sin a cos ft — cos a sin /3 (why ?) . 

Also cos (a — 0)= cosacos(— ft)— since sin (— 0) 

= cos a cos ft + sin a sin ft. 

Finally, 

+ r a. /Q\ _ sin (a ± £) _ si n a cos ft ± cos a sin 

tan ( a ± p ) — — ~ : : — 

cos (a ± 0) cos a cos ft T sin a sm ft 

sin a cos /3 cos a sin /3 

cos a cos /3 cos a cos /3 tan a ± tan /3 



cos a cos /3 sin a sin ft 1 q: tan cc tan ft 
cos a cos ft cos a cos /3 



EXAMPLES. 59 

ORAL WORK. 

By the above formulae develop : 

1. sin (2 A + 3 B) . 7. sin 90° = sin (45° + 45°). 

2. cos (90° -B). 8. cos 90°. 

3. tan (45° + <£). 9. tan 90°. 

4. sin 2 A = sin (J. + A). 10. sin (90° + j3 + y). 

5. cos 2 0. 11. cos (270° - m - n). 

6. tan (180° + C). 12. tan (90° + m + n). 

Ex. 1. Find sin 75°. 

sin 75° = sin (45° + 30°) = sin 45° cos 30° + cos 45° sin 30° 

V2 2 V2 2 2V2 

2. Find tan 15°. 

tan 45° - tan 30° 



tan 15° = tan (45° - 30°) = 



1 +tan45°tan30 c 



1-- i. 

= ^ = ^3-l = 2 - V3 = 0.2679. 

1 + J_ V3 + 1 
V3 

3. Prove sin 3 A - cos 3 A = 2. 
sin A cos A 

Combining sm3 ^ cos^4-cos3J. sin^4 _ sin (3 A - A) 
sin A cos A sin A cos A 

_ sin 2 A _ sin (A + ^4) _ sin A cos A + cos A sin A _q 
sin A cos A sin ^4 cos -4 sin A cos .4 



4. Prove tan -1 a + tan -1 b = tan -1 — 



l-a& 

Let a = tan- 1 a, B = tan -1 b, y = tan -1 -^Jl — 

1 — a& 

Hence, tan a = a, tan B = b, tan y = — — - — 

1 — ab 

Then a + /? = y, and hence tan (a + (3) = tan y. 

Expanding, tan <* + tan /? = tan 

1 — tan a tan /3 

Substituting, a + h = a + h . 

5 1 - ab 1 - ab 



60 PLANE TRIGONOMETRY. 

EXAMPLES. 

1. Find cos 15°, tan 75°. 

2. Prove cot (a ± 0) = ™* a cot 0*1 . 

v ^ J cot /? ± cot a 

3. Prove geometrically sin (a + (3) = sin a cos (3 + cos a sin /J, 

and cos (a + /?) = cos ct cos /3 — sin a sin /?, 
given («) ce acute, /? obtuse ; 

(b) a, (3, obtuse ; 

(c) a, /?, either, or both, negative angles. 

4. Prove geometrically tan (a + (3) = tan a + tan P . 

5 J v A-y l_tanatan/? 

Verify the formula by assigning values to a and (3, and finding the 
values of the functions from the tables of natural tangents. 

5. Prove cos (a + (3) cos (a — (3) = cos 2 a — sin 2 (3. 

6. Show that tan a + tan (3 = sm ^ + ^ - 

cos a cos p 

7. Given tan a = \, tan /? = f , find sin (a + /3) 

8. Given sin 280° = s, find sin 170°. 

9. If a = 67° 22', = 128° 40', by use of the tables of natural func- 
tions verify the formulae on page 56. 

10. Prove tan- 1 + _ = tan- 1 VaT + tan^Va. 
1 —Vax 

tan-iVB. 



11. 


Prove tan- 12 x h + tan" 1 2 & ~ x 
by/3 xV3 


12. 


Prove sec -1 = sin -1 -• 




Va 2 - x 2 a 



13. If a + (3 = to, prove cos 2 a + cos 2 /? — 2 cos a cos /? cos <o = sin 2 <u. 

14. Solve \ sin = 1 — cos 0. 

15. Prove sin (A + -B) cos A — cos (yl + B) sin J. = sin B. 

16. Prove cos {A + 5) cos (J. -B) + sin (^ + B) sin(^i - £) = cos 2 5. 

17. Prove sin (2 a - (3) cos (a -2/3) 

- cos (2 a- /3) sin (a - 2 @) = sin (a + j3). 

18. Prove sin(n — l)ot cos(w + l)« + cos(n-l)a sin(?i + l)ot = sin2 na. 

19. Prove sin (135° - 0) + cos (135° + 6) = 0. 



ADDITION— SUBTRACTION FORMULAE. 61 

nr\ • "D 1 t 9 4- 9 O COS 2 /? — Sill' 2 « 

20. Prove 1 — tan 2 a tan 2 ft = *=■ • 

cos 2 a cos 2 (3 

„, -d tan a + tan j8 . . 

21. Prove ■ £- = tan a tan «. 

cot ct + cot p 

22. tan 2 f * - «") = 1 ~ 2 sin « cos « 

\4 / 1 + 2 sin a cos « 

51. The following formulae are very important and should 
be carefully memorized. They enable us to change sums 
and differences to products, i.e. to displace terms by factors. 

sin + sin 4> = 2 sin— —^ cos ~ % 

sine - sin<|) = 2 cos "^ * sin ~ **% 

« 6+6 0—6 
COS0 + COS<f> = 2 COS o COS— ^y 

cos 9 - cos<j> = - 2 sin ^ sin — ^* 

Since sin (« + /3) = sin « cos /3 + cos a sin /3, 

and sin (« — j3) = sin a cos /3 — cos a sin /3, 

then sin (« -j- y8) + sin (a — /3) = 2 sin « cos /3, (1) 

and sin (« + /3) — sin (a — /3) = 2 cos a sin /3. (2) 

Also since cos (« + /3) = cos a cos /3 — sin a sin /3, 

and cos (a — /3) = cos « cos /3 + sin a sin /3, 

then cos (« + /3) + cos (« — /3) = 2 cos « cos /3, (3) 

and cos (a + /3) — cos (a — /3) = — 2 sin « sin /3. (4) 

Put (* + £ = 6 

and a — ft = <j) 

2a = + 0, and a = i±A, 



2/3 = 0-<£, and/3 = ^-=-^. 



Substituting in (1), (2), (3), (4), we have the above 
formulae. 



62 PLANE TRIGONOMETRY. 



EXAMPLES. 

!. Prove sin2^sin^ tan 3^ 
cos 2 $ + cos 2 

By formulae of last article the first member becomes 

. 30 6 

2 sin — cos - 

2 2 - t<m 3 

2cos^cos^ an 2 ' 
2 2 

sin a 4- 2 sin 3 # + sin 5 a sin 3 a 



2. Prove 



sin 3 a + 2 sin 5 a + sin 7 a sin 5 a 

(sin a + sin 5 a) + 2 sin 3 a _ 2 sin 3 a cos 2 & + 2 sin 3 ct 
(sin 3 a + sin 7 a) + 2 sin 5 a 2 sin 5 a cos 2 a + 2 sin 5 a 

_ (cos 2 rc + 1) sin 3 a _ sin 3 a 
(cos 2 a + 1) sin 5 a sin 5 cc 

3. Prove sin (4 ^ - 2 i?) + sin (4 5 - 2 .4) = 

cos (4,4 -2 5)4-008(45-2^) v y 

. 4i-25 + 4B-2i irt 4i-25-45 + 2i 

w Sill. COS ■ 

2 2 

±A-2B + ±B-2A 4=A-2B-4B + 2A 

2 COS ■ COS ! 

2 2 

= sm(.4+i?) = (i B) 

cos (.4 +5) v y 

4. Prove sin 50° - sin 70° + sin 10° = 0. 

2 cos 50 ° ± 70 ° sin 50 ° ~ 70 ° = 2 cos 60° sin ( - 10°) = - sin 10°. 

5. Piwe cos2ctcos3ct - cos2ctcos7a + cos ^ cosl0f)t ^cot6acot5^ 

sin 4 a sin 3 a — sin 2 a sin 5 « + sin 4 # sin 7 a 

By (3) and (4), p. 61, 

cos 5 a + cos a — cos 9 a — cos 5 a + cos 1 1 a + cos 9 a 
cos a — cos 7 ct — cos 3 a + cos 7 « + cos 3 a — cos 11 « 

cos a + cos 11 a 2 cos 6 a cos 5 « ,. ft „ ,. K „ 

= ■ = — ; ; = cot d a cot 5 a. 

cos a — cos 11 a 2 sin 6 a sin 5 a 

ORAL WORK. 
By the formulae of Art. 51 transform : 

6. cos 5 a + cos a. 8. 2 sin 3 6 cos 0. 

7. cos a — cos 5 a. 9. sin 2 cc — sin 4 a. 



FUNCTIONS OF THE DOUBLE ANGLE. 63 

10. cos 9 6 cos 2 0. 16. cos (30° +2cf>) sin (30° - <f>). 

11. sin + sin |. 17. sin (2 r + s) + sin (2 r - s). 

12. sin 75° sin 15°. 18 - cos ( 2 £ - «) - cos 3 a. 

13. 0087^-0082^. 19. sin 36° + sin 54°. 

14. cos(2p + 3o)sin(2o — 3o). ^ 

v /'-i <*; v r hj 2Q cos 60° + cos 20°. 

15. sin— -sin-- 

2 2 21. sin 30° + cos 30°. 

Prove: 22. sin a + sin g = tan «+£ cot gLl£ 

sin a — sin g 2 2 

23> cos « + cos g = cot «±£ cot £zig. 

cos g — cos a 2 2 

24 sin a ; + siny :=tan £+^ j 
cos a; + cosy 2 

25> sin g- sin y = _ ^ x_±y^ 
cos x — cos y 2 

26. cos 55° + sin 25° = sin 85°. 

sin B -f sin 2 B + sin 3 i> 



Simplify : 27. 



28. 



cos B -f cos 2 J5 + cos 3 f> 

sin C - sin 4 C + sin 7 C - sin 10 g 
cos C — cos 4 C + cos 7 C — cos 10 C 



52. Functions of an angle in terms of those of the half angle. 
If in sin (« + /3) = sin a cos /3 + cos a sin 0, a = j3, 
then sin (« + «)= sin 2 a = 2 sin a cos a. 
In like manner 

cos (« + «)= cos 2 a = cos 2 a - sin 3 a 
= 2 cos 2 a - 1 
= 1-2 sin 2 a ; 

and tan2a= 2tana 



1 - tans a 



64 PLANE TRIGONOMETRY. 

ORAL WORK. 

Ex. Express in terms of functions of half the given angles : 

1. sin 4 a. 4. cos a;. 6. sin(2jt? — q). 

2. cos3i?. . p 7. cos (30° + 2 <£). 

5 . sin *-— c 

3. tan 5 t. 2 s. sin (x + y). 

9. From the functions of 30° find those of 60° ; from the functions of 
45°, those of 90°. 

53. Functions of an angle in terms of those of twice the angle. 

By Art. 52, cos a = 1 - 2 sin 2 2 = 2 cos 2 - - 1. 

J 2 2 

.*. 2 sin 2 - = 1 — cos «, and 2 cos 2 - = 1 -f cos a. 

a ^ /l -cos a -a , . /l + < 



Sill 



cos a 
2 



a 

sin 



a 2 1 - cos a 

tan- = = ±\- 

2 « * 1 + cos a 



cos 2 

Explain the significance of the ± sign before the radicals. 
Express in terms of the double angle the functions of 
120°; 50°; 90°, with proper signs prefixed. 

Ex. 1. Express in terms of functions of twice the given angles each 
of the functions in Examples 1-8 above. 

2. From the functions of 45° find those of 22° 30' ; from the functions 
of 36°, those of 18° (see tables of natural functions). 

3. Find the corresponding functions of twice and of half each of the 
following angles, and verify results by the tables of natural functions : 

Given sin 26° 42' = 0.4493, 

tan 62° 24' = 1.9128, 

cos 21° 34' = 0.9300. 



->fe 



V/ X ~ cos x = -• 5. 2tan- 1 z=tan-i-^ 

cos X 2 1 — 5 



EXAMPLES. 



65 



6. If A, B, C are angles of a triangle, prove 

ABC 
sin A + sin C + sini? = 4 cos — sin — sin — • 

Ji £ £ 

7. If cos 2 a + cos 2 2 a + cos 2 3 a = 1, then 

cos a cos 2 « cos 3 a = 0. 

8. Prove cot A — cot 2 A = esc 2 A. 



tan 



9. Prove 



4> 



10. 



tan 

tana 



(M) i 



1 — tan 



41 



1 + tan x. 

9 



sin <jy 



tan (« + <£) sin (2 « + <£) + sin cf> 



11. If y = tan- 1 V l + x ' 2 + V l x ' 2 , prove x 2 = sin 2^. 

Vl + x 1 — a/1 — a; 2 

12. Prove tan- 1 Vl + j5z 1 | tan- 1 

x 

13. Uy = sin- 1 



= 5 tan- 1 x. 



1 -x 2 
prove z — tan y. 



a/1 + a- 2 

14. Prove cos 2 a + cos 2 /? — 1 = cos (ct + j3) cos (a — (3). 

15. Prove V(cos a - cos fi) 2 + (sin a - sin /3) 2 = 2 sin f * ~^ > 



COS" 



16. Prove sin" 1 -J— ^— = tan- 1 a/- = I 

^a+x * a 2 a + x 

17. Prove cos 2 - cos 2 <£ = sin (<£ + 0) sin (<£ - 6). 

18. Prove tan A + tan (4 + 120°) + tan {A - 120°) = 3 tan 3 A, 

19. Prove tan a — tan - = tan - sec a. 

2 2 

20. 3 tan- 1 a = tan- 1 8 a ~ a8 - 

1 -3a 2 

21. cos 2 3 J (tan 2 3 A - tan 2 .4) = 8 sin 2 .4 cos 2 A. 

22. 1 + cos 2 (.4 - B) cos 2 5 = cos 2 A + cos 2 (J. -25). 
2 (9 - sec 



23. cot 2 (E + ^ = 2cscA 
U 2/ 2 esc 2 



+ sec 



66 PLANE TRIGONOMETRY. 

TRIGONOMETRIC EQUATIONS AND IDENTITIES. 

54. Identities. It was shown in Chapter I that 

sin 2 6 + cos 2 6 = 1 

is true for all values of 0, and in Chapter VI, that 

sin (ct + /3) = sin a cos (3 + cos a sin /3 

is true for all values of a and ft. It may be shown that 

sin 2 A , A 

- — — = tan A 

1 + cos 2 A 

is true for all values of A, thus : 

sin 2 A 2 sin A cos J. (by trigonometric transf orma- 

1 + cos 2 A ~ 1 + 2 cos 2 A - 1 tion) 

= — — - (by algebraic transformation ) 

cos A 

= tan^L (by trigonometric transformation). 

Such expressions are called trigonometric identities. They 
are true for all values of the angles involved. 

55. Equations. The expression 

2 cos 2 a — 3 cos a + 1 = 

is true for but two values of cos a, viz. cos a= J and 1, i.e. 
the expression is true for a = 0°, 60°, 300°, and for no other 
positive angles less than 360°. Such expressions are called 
trigonometric equations. They are true only for particular 
values of the angles involved. 

56. Method of attack. The transformations necessary at 
any step in the proof of identities, or the solution of equa- 
tions, are either trigonometric, or algebraic; i.e. in prov- 
ing an identity, or solving an equation, the student must 
choose at each step to apply either some principles of algebra, 
or some trigonometric relations. If at any step no algebraic 
operation seems advantageous, then usually the expression 



METHOD OF ATTACK. 67 

should be simplified by endeavoring to state the different 
functions involved in terms of a single function of the angle, 
or if there are multiple angles, to reduce all to functions of a 
single angle. 

' Algebraic 



Transformations 



I Trigonometric, f Single function 
to change to a I Single angle 



No other transformations are needed, and the student will 
be greatly assisted by remembering that the ready solution 
of a trigonometric problem consists in wisely choosing at 
each step between the possible algebraic and trigonometric 
transformations. Problems involving trigonometric func- 
tions will in general be simplified by expressing them entirely 
in terms of sine and cosine. 





EXAMPLES, 


1. Prove 


sin 3 A cos 3 A _ 




sin A cos A 


-d t i sin 3 A 
By algebra, — — - - 


cos 3 A sin 3 A cos A — cos 3 A sin A 


J & sin A 


cos A sin A cos A 


by trigonometry, 


_ sin (3 A - A ) _ sin 2 A 
sin A cos A sin A cos A 




_ 2 sin A cos vl _ 
sin J. cos J. 


Or, by trigonometry, 




sin 3 A cos 3 A 


3 sin A — 4 sin 3 A 4 cos 3 ,4 — 3 cos A 


sin A cos A 


sin .4 cos A 


by algebra, 


= 3 - 4sin 2 yl - 4 cos 2 .4 + 3 




= 6 - 4(sin 2 .4 + cos 2 .4)= 2. 


2. Prove 


sec 8 - 1 tan 8 
sec 4 6 - 1 tan 2 



No algebraic operation simplifies. Two trigonometric changes are 
needed. 1. To change the functions to a single function, sine or cosine. 
2. To change the angles to a single angle, 8 A, 4: A, or 2 A. 



68 PLANE TRIGONOMETRY. 

By trigonometry and algebra, 

1 - cos 8 sin 8 
cos 8 cos 8 



1 - cos 4 sin 2 ' 
cos 4 cos 2 

by algebra, cos 4 0(1 - cos 8 fl) = sin8flcos2fl . 

1 — cos 4 sin 2 

by trigonometry, 

cos 4 0(1 -1 + 2 sin 2 4 0) = 2 sin 4 cos 4 cos 2 . 
l-l + 2sin 2 20 sin20 ' 

by algebra, ^|i=2cos20; 

sm 2 

and sin 4 = 2 sin 2 cos 2 0, 

which is a trigonometric identity. 

3. Solve 2 cos 2 + 3 sin = 0. 

By trigonometry, 2(1 - sin 2 0) + 3 sin = 0, 
a quadratic equation in sin 0. 

By algebra, 2 sin 2 - 3 sin - 2 = 0, 

and (sin 0-2) (2 sin + 1) = 0. 

.\ sin = 2, or — i. Verify. 

The value 2 must be rejected. Why? 

.-. = 210°, and 330° are the only positive values less than 360° that 
satisfy the equation. 

4. Solve sec - tan = 2. 

Here tan = — 0.75, .-. from the tables of natural functions, 

4 143° V 48", or 323° 7' 48". 
Find sec 0, and verify. 

5. Solve 2 sin sin 3 - sin 2 2 = 0. 

By trigonometry, cos 2 — cos 4 — sin 2 2 = 0, 
also cos 2 - cos 2 2 + sin 2 2 - sin 2 20 = 0. 

By algebra, cos 2 0(1 - cos 2 0) = 0. 

.-. cos 2 = or 1, 
and 2 = 90°, 270°, 0°, or 360°, 

whence = 45°, 135°, 0°, or 180°. Verify. 



TRIGONOMETRIC EQUATIONS. 69 

Or, by trigonometry, 

2 sin 0(3 sin 0-4 sin 3 0) - 4 sin 2 cos 2 = ; 
by trigonometry and algebra, 

6 sin 2 0-8 sin 4 (9-4 sin 2 (9 + 4 sin 4 = 0; 
by algebra, 2 sin 2 0-4 sin 4 = 0, 

and 2 sin 2 0(1 -2sin 2 0) = O. 

.-. sin = 0, or ± VJ, 
and = 0°, 180°, 45°, 135°, 225°, or 315°. 

The last two values do not appear in the first solution, because only 
angles less than 360° are considered, and the solution there gave values 
of 2 0, which in the last two cases would be 450° and 630°. 

Solve : 6. tan = cot 0. 8. 2 cos 2 - 2 sin 0=1. 

7. sin 2 + cos = 1. 9. sin 2 cos = sin 0, 

Prove : 10. 2 cot 2 A = cot A - tan A . 

11. cos 2 x + cos 2 ?/ = 2 cos (a; + y) cos (x — #). 

12. (cos « + sin a)- = 1 + sin 2 a. 

57. Simultaneous trigonometric equations. 

13. Solve cos (x + y) + cos (a; - ?/) = 2, 

sin - + sin ^ = 0. 
2 2 

By trigonometry, 

cos x cos «/ — sin a; sin ?/ + cos x cos ?/ + sin x sin y = 2, 
so that cos x cos ?/ = 1 ; 



also, 



^ i -_co ii;+ ^l^os 1 = 0j 



and ••• cos x = cos y. 

Substituting, cos 2 x = 1, 

cos 2' =± 1. 
.-. x = 0°, or 180°, 
and y = x = 0°, or 180°. Verify. 



70 PLANE TRIGONOMETRY 

14. Solve for R and F. 

W — Fsini — R cos i = 0, 

W + Fcos i - R sin i = 0. 
To eliminate F, 

W cos i — F sin i cos i — R cos 2 i = 0, 

W sin £ + i^cos i sin i — R sin 2 £ = 0. 
Adding, TF(sin £ + cos i) — i?(sin 2 i + cos 2 i) = 0. 

.•. R = TF(sin £ + cos z). 
Substituting, W — Fsini — W(sin i + cos i)cos i = 

rr _ TT — Ty(sin i + cos i) cos i 
sin i 
If TT = 3 tons, and i = 22° 30', compute F and i2. 

£ = 3(0.3827 + 0.9239)= 3.9198. 

F = 3 - 3(0.3827 + 0.9239)0.9239 = _ x 624 

0.3827 
Solve : 

15. 472 cot - 263 cot <f> = 490, 307 cot - 379 cot <j> = 0. 

16. sin 2 a; + 1 = cos # + 2 sin #. 

17. cos 2 + sin = 1. 

18. If 2A(cos 2 0-sin 2 0)-2asin0cos0 + 2&sin0cos0=O, prove 
= i tan" 1 2 



a — 5 
Prove : 

19. tan ?/ = (1 + sec ?/) tan ^- 

20. 2 cot- 1 a; = esc" 1 1 + — • 

2;r 

21. sin(<£ + 45°) + sin (<£ + 135°) = V2 cos <£. 
22 cos v + cos 3 v _ 1 



cos 3 v + cos 5 v 2 cos 2 y — sec 2 » 

23. cos 3 x — sin 3 # = (cos x + sin #) (1 — 2 sin 2 #). 
Solve : 

24. sin 2 + sin = cos 2 + cos 0. 

25. 4 cos(0 + 60°) - V2 = V6 - 4 cos (0 + 30°). 

26. cot 2 = tan 0-1. 

27. cos + cos 2 + cos 3 = 0. 



TRIGONOMETRIC EQUATIONS. 71 

28. sin 2 a: + V3 cos 2x = 1. 

29. 3 tan 2 jo + 8 cos 2 ^ = 7. 

30. Determine for what relative values of P and W the following 
equation is true : 

CO s 2 ^--^-cos^--= 0. 
2 W 2 2 

31. Compute iVfrom the equation iV+ — cosct sin a — TFcos« = 7 

o o 

when W = 2000 pounds and ex. satisfies the equation 2 sin a = 1 + cos a. 

32. sin 6 — tan <£(cos $ + sin 0) = cos 6, sin — tan <f> cos = 1. 
Prove : 

33. cot(t + 15°) - tan (7 - 15°) = * co f 2 * . 

2sm2< + l 

34. sin -1 f — sin -1 T 5 3 = sin -1 -|f. 



35. tan(j + £Wl + 4^- 

\4 2 / *1 — sm o> 

36. 2 sin" 1 1 = cos" 1 -|. 

37. If sin A is a geometric mean between sin B and cos B, prove 
cos 2 ,4 =2sin(45 - B) cos (45 + B). 

38. Prove sin(« -f (3 + y) = sin ct cos (3 cos y + cos a sin /3 cos y 

+ cos « cos (3 sin y — sin a sin /} sin y. 
Also find cos(a -f (3 + y). 

^rt -r, , K , , \ tan « + tan /3 + tan v — tan « tan B tan v 

39. Prove tan (a + B + y) = ! *=-: ^ ^ '. 

1 — tan a tan (3 — tan (3 tan y — tan y tan a 

If a, (3, and y are angles of a triangle, prove 

40. tan « + tan /? + tan y = tan « tan /3 tan y. 

41. cot - + cot £ + cot 2 = cot - cot £ cot 2. 

2 2 2 2 2 2 

If« + j8 + y = 90°, prove 

42. tan a tan f3 + tan /3 tan y + tan y tan a = 1. 
Prove : 

43. sin n« = 2 sin (n — 1) a cos « — sin (n — 2)«. 

44. cos na = 2 cos (n — 1) « cos a — cos (n — 2)a. 

.j- , tan fn — 1) a + tan ot 

45. tan na = * ; - — 

1 — tan (n — 1) a tan a 



CHAPTER VII. 

TRIANGLES. 

58. In geometry it has been shown that a triangle is 
determined, except in the ambiguous case, if there are given 
any three independent parts, as follows : 

I. Two angles and a side. 
II. Two sides and an angle, 
(a) the angle being included by the given sides, 
(6) the angle being opposite one of the given sides (am- 
biguous case). 
III. Three sides. 

The angles of a triangle are not three independent parts, since they 
are connected by the relation A + B + C = 180°. 

The three angles of a triangle will be designated A, B, O, 
the sides opposite, a, b, c. 

But the principles of geometry do not enable us to compute 
the unknown parts. This is accomplished by the following 
laws of trigonometry : 

sin A sin B sin 



I. Law of Sines, 



a b c 



II. Law of Tangents, an g \ A — - = - — -, etc. 

J U tan I (A + B) a + V 

2 _i_ c 2 _ a 2 

III. Law of Cosines, cos A = — -^— , etc. 

2 be 

59. Law of Sines. In any triangle the sides are propor- 
tional to the sines of the angles opposite. 

Let ABO be any triangle, p the perpendicular from B 
on b. In I (Fig. 34), C is an acute, in II, an obtuse, in III, 

72 



LAW OF SIXES — OF TANGENTS. 



73 



a right angle. The demonstration applies to each triangle, 
but in II, sin ACB= sin BOB (why?); in III, sin (7=1 
(why?). 

B 




b D C A b C D A b 

I. II. III. 

Fig. 34. 



Now 



sin A = 



p 



sinC=^ 



Equating values of p, 



or, 



.*. jp = c sin -4. 

.-. p = a sin Q. 

c sin A = a sin C, 
sin A sin C 



By dropping a perpendicular from A, or G, on a, or c, show 
that 



whence 



sin B 
b 

sin J. 



sin C sin A 
, or 



sinB 



c 

sin B 



a 
sin 6' 



60. Law of Tangents. The tangent of half the difference 
of two angles of a triangle is to the tangent of half their sum, 
as the difference of the sides opposite is to their sum. 

a sin A 
b 



By Art. 59, 



sinB 
By composition and division, 

a — b_ sin A — sin B _ 2 cos | (A + i?) sin j - (A — B) 
a + b~ sinA + sin B ~" 2 sin 1 ( J. + i?) cos 1 (A — .B) 

= tan^(^-jB) . 
tan I^Z+iT)' 



or, 



tan j- ( J, — i?) _ a — b 
tan 1 ( J. + B) ~ a + 5' 



74 



PLANE TRIGONOMETRY. 



61. Law of Cosines. The cosine of any angle of a triangle 
is equal to the quotient of the sum of the squares of the adjacent 
sides less the square of the opposite side, divided by twice the 
product of the adjacent sides. 




b D C A b C D A b 

I. II. III. 

Fig. 34. 



In each figure a 2 =p 2 + DC 2 

= e 2 -AD 2 + (b-ADy 
(in Fig. 34, II, DC is negative; in III, zero) 

= c 2 _ AI) 2 + b*-2b-AD + AD 2 

= b 2 + c 2 -2b-AD. 



But 














AD = 


= c cos 


A, 


.'. a 2 = 

cos A = 


b 2 + c 2 - 

b 2 + c 2 - 
2 be 


2 be cos 
a 2 


A; 


Prove 


that 




cosB = 


a 2 + c 2 - 
2 ac 


b 2 




and 






cos C = 


a 2 + b 2 - 


-c* 





2 ah 

62. Though these formulae may be used for the solution 
of the triangle, they are not adapted to the use of loga- 
rithms (why?). Hence we derive the following: 

Since cos A = 2 cos 2 4-1 = 1-2 sin 2 ^, 

2 2' 

we have 

oA . A 

2 cos 2 — = 1 + cos A, and 2 sin 2 — = 1 — cos A, 

A A 



LAW OF COSINES. 75 

From the latter 

o . oA -, b 2 + c 2 -a 2 2be-b 2 -c 2 + a 2 

2S111 2 =1 - 2be = 2be 

= a 2 _ ( h _ c y (a _ i + g ) (a + b _ ^ 
2 be 2 be 

Let a + b + c = 2s, then a + b — c=a + b + e — 2e = 2s — 2c, 
i.e. a + b — c = 2 (s — c). 

In like manner, a — b + c = 2 (s — b~). 
— a + b + c=2(s-a). 

Substituting, 2 sin 2 — = — ^ ^' ^ ~" g A 

2 2 oc 

.-. sin— = V^ f^ ^ 

Show that sin — = ? 

also s ^ n ~r = ? 

From 2 cos 2 — = 1 + cos A, 



show that cos — = \ -^— : — — 



2 * be 



also cos — = ? 

and cos — = ? 

2 

Also derive the formulae 



2 y s(s — a) 

tan- = ? 

2 



76 PLANE TRIGONOMETRY. 

63. Area of the triangle. In the figures of Art. 59 the 

area of the triangle ABO '== A = \fb. 

But p=esinA. . \ A = \be sin A. (i) 

csinB 



Again, by law of sines, b 
Substituting, A = 

& Sill \J 

(why?). (ii) 



sin C 

c 2 sin^L sin B 

2 sin 

e 2 sin A sin B 
2sin(A + B) 



A A 

Finally, since sin A = 2 sin — cos — , we have from (i) 

A A 

a ii o • A A 7 ^ls(s — a)(s— b)(s — c) 

A = I be - 2 sin— cos — = bc\\-± *f -^ ^ 

2 2 2 x be • be 



or A = Vs (s — «) (s — 6) (s — e) . (iii) 

Find A ; (1) Given a = 10, b = 12, C = 45°. 

(2) Given a = 4, 6 = 5, c = 6. 

(3) Given a = 2, £ = 45°, C = 60°. 

SOLUTION OF TRIANGLES. 

64. For the solution of triangles we have the following 
f ormulge, which should be carefully memorized : 

j sin A _ sin B _ sin C 
a b c 

II. tan I (A- B) = ^=\ tan I (A + B). 
a + b 

III. sini = J ( s - h l(^ or ^-JSSS, 

A y be A y be 



or tan|=V (S " 6)(s " c) - 



2 ^ s(s - a) 

IV. A = | be sin ^ = ^^hi 2? = ^ _ a) (8 - 6) (* - c). 
2 2 sin (A + .B) 



SOLUTION OF TRIANGLES. 77 

Which of the above formulae shall be used in the solution 
of a given triangle must be determined by examining the 
parts known, as will appear in Art. 69. It is always pos- 
sible to express each of the unknown parts in terms of three 
known parts. 

In solving triangles such as Case I, Art. 58, the law of 
sines applies; for, if the given side is not opposite either 
given angle, the third angle of the triangle is found from 
the relation A + B + C= 180°, and then three of the four 

quantities in = — - — being known, the solution gives 

a b 

the fourth. 

In Case II (b~) the law of sines applies, but in II (a) two 

only of the four quantities in = are known. 

a b 

Therefore, we resort to the formula 

tan J ( A - B) = ^—^ tan £ (A + -#), 

in which all the factors of the second member are known. 
In Case III, tan^ = ^MKM i s clearly applicable, 

— i S \^S — CI J 

A A 

and is preferred to the formulae for sin — and cos — ; for, 

first, it is more accurate since tangent varies in magnitude 
from to oc, while sine and cosine lie between and 1. 

(See Art. 27, 5.) 

Let the student satisfy himself on this point by finding, correct to 
seconds, the angle whose logarithmic sine is 9.99992, and whose loga- 
rithmic tangent is 1.71668. Does the first determine the angle ? Does 
the second? 

And, second, it is more convenient, since in the complete 

A 

solution of the triangle by sin — six logarithms must be taken 

A A 

from the table, by cos — seven, and by tan — but four. 

A '— 

The right triangle may be solved as a special case by the 
law of sines, since sin 0=1. 



78 



PLANE TRIGONOMETRY. 



65. Ambiguous case. In geometry it was proved that a 
triangle having two sides and an angle opposite one of them 
of given magnitude is not always determined. The marks 
of the undetermined or ambiguous triangle are : 

1. The parts given are two sides and an angle opposite one. 

2. The given angle is acute. 

3. The side opposite this angle is less than the other given 
side. 

When these marks are all present, the number of solutions 
must be tested in one of two ways : 

(a) From the figure it is apparent that there will be no 
solution when the side opposite is less than the perpendicular 
p ; one solution when side a equals p ; and two solutions when 
a is greater than p. 




A b C A b C A b C & 

No Solution. One Solution. Two Solutions. 

Fig. 35. 



And since sin A 



V 



it follows that there will be no solu- 



tion, one solution, two solutions, according as sin A 



>a 
<~c 



(5) A good test is found in solving by means of loga- 
rithms ; and there will be no solutions, one solution, two solu- 
tions, according as log sin C proves to be impossible, zero, 
possible, i.e. as log sin C is positive, zero, or negative. This 
results from the fact that sine cannot be greater than unity, 
whence log sine must have a negative characteristic, or be 
zero. 

66. In computations time and accuracy assume more than 
usual importance. Time will be saved by an orderly arrange- 
ment of the formulse for the complete solution, before open- 
ing the book of logarithms, thus : 



SOLUTION OF TRIANGLES. 79 



c 



Given A, B, a. £ 


Solve complet 


ely. 


= 180°-(J.+J5), 


7 a sin i? 
o= .. . » 
sin ^4 


flsinC A i i • /7 

e = — » A = i <^o sin 6. 

sin J. 2 


180° 
A+B = 


log a = 
log sin i>> = 


log a = 
log sin C = 


.-. C = 


colog sin A = 
log & = 


colog sin ^4 = 




log c = 




.-. b = 


.-. c = 

Check : 


log a = 

logb = 

log sin C = 

colog 2 = 


log (s - b) = 

log (s - c) = 

colog s = 

colog (s — a) = 


log A = 




2) 


.-.A = 




log tan — = 



.-. A = 

67. Accuracy must be secured by checks on the work at 
every step ; e.g. in adding columns of logarithms, first add 
up, and then check by adding down. Too much care can- 
not be given to verification in the simple operations of 
addition, subtraction, multiplication, and division. A final 
check should be made by using other formulae involving the 
parts in a different way, as in the check above. As far as 
possible the parts originally given should be used through- 
out in the solution, so that an error in computing one part 
may not affect later computations. 

68. The formulae should always be solved for the unknown 
part before using, and it should be noted whether the solu- 
tion gives one value, or more than one, for each part ; e.g. 
the same value of sin B belongs to two supplementary angles, 
one or both of which may be possible, as in the ambiguous 
case. 

69. Write formulae for the complete solution of the fol- 
lowing triangles, showing whether you find no solution, one 
solution, two or more solutions, in each case, with reasons for 
your conclusion : 



80 



PLANE TRIGONOMETRY. 





a 


b 


c 


A 


B 


C 


1. 








81° 26' 28" 


44° 11' 20" 


54° 22' 12" 


2. 




78.54 




63° 18' 20" 




41° 30' 18" 


3. 




135.82 


26.89 


53° 28' 30" 






4. 


0.75 


0.85 


0.95 








5. 


243 




562 






36° 15' 40" 


6. 




38.75 


25.92 






63° 50' 10" 


7. 


0.058 






78° 15' 


33° 46' 




8. 


2986 




1493 






30° 


9. 




48 


50 




26° 15' 





MODEL SOLUTIONS. 



1. Given a = 0.785, b = 0.85, c = 0.633. Solve completely. 



tan 4 =V<'-;X'7 C >, tan f =^-^'~ e \ tan CJ-'X-V. 
2 y s(s — a) 2 ' s(s — b) 2 * s(s — c) 



Check: A + B + C = 180°. A = Vs(s - a)(s - b)(s - c). 



a = 0.735 
b = 0.85 
c = 0.633 



2 )2.268 

s = 1.134 

s-a = 0.349 

s -b = 0.284 

s - c = 0.501 

Check : 

A = 61° 53' 38" 

B= 72° 46' 4" 

C= 45° 20' 20" 



180° 0' 2" 



log (s -b)= 9.45332 

log (s-c)= 9.69984 

cologs= 9.94539 

colog (>•-«)= 0.45717 

2 )19.55572 
log tan %A= 9.77786 

\A =30° 56' 49" 
A = 61° 53' 38" 

log (s - a) = 9.54283 

log (s -b)= 9.45332 

cologs= 9.94539 

colog(s-c)= 0.30016 

2 )19.24170 
logtaniC= 9.62085 

i C = 22° 40' 10" 
C = 45° 20' 20" 



log (s - a) 
log (6> - c) 

colog s 
colog (5 — b) 



= 9.54283 

= 9.69984 

= 9.94539 

= 0.54668 



2 )19.73474 
log tan \ B = 9.86737 
i£ = 36°23'2" 

= 72° 46' 4" 



B 

logs 
log (s - a) 
log - 6) 
log (5 - c) 



= 0.05461 

= 9.54283 

= 9.45332 

= 9.69984 

2)18.75060 
= 9.37530 



lOg A : 

A = 0.2373 



Solve :(1) Given a = 30, b = 40, c = 50. 

(2) Given a = 2159, b = 1431.6, c = 914.8. 

(3) Given a = 78.54, b = 32.56, c = 48.9. 



SOLUTION OF TRIANGLES. 



81 



2. Given 4 =57° 23' 12", C = 68°15'30' 



c sin A 

a = 

sin C 

B = 180°-(A + C) 
= 51° 21' 18". 



b = 



c sin B 



sin C ' 
Check: tan ^^4 



832.56. Solve completely. 



A = * 6c sin J. . 



V^ 



&)(s-c) 



c = 2.92042 
.92548 
colog sin C = 0.03204 



log sin A = 



log a = 2.87794 
a= 754.98 



log c = 2.92042 

log sin £ = 9.90990 

colog sin C = 0.03204 

log b = 2.86236 
6= 728.38 



A = 



6 - (*• — a) 

log 6= 2.86236 

logc= 2.92042 

log sin yl = 9.92548 

log 2 A = 5.70826 
510811 



= 255405.5 



Check: a 
b 



754.98 
728.38 
c= 832.56 



s — a 

s- b 
s — c 



402.98 
429.58 
325.40 



2 )2315.92 
s = 1157.96 



log (*-&)= 2.63304 

log(s-c) = 2.51242 

colog 5= 6.93634 

colog (s — a) = 7.39471 



log tan 



2 )19.47651 
\A- 9.73826 
\A- 28° 41' 38" 
A = 57° 23' 16" 



Solve : 

(1) Given a = 215.73, B = 92° 15', C = 28° 14'. 

(2) Given b = 0.827, A = 78° 14' 20", B = 63° 42' 30". 

(3) Given b = 7.54, c = 6.93, i* = 54° 28' 40". 

3. Given a = 25.384, c = 52.925, B = 28° 32' 20". Solve completely. 
(Why not use the same formulae as in Example 1, or 2?) 
C - A c — a ,. C+.4 , c sin 5 



„ - a , C + A , 

tan = tan — - — , b 

2 c + a 2 

180° - 5 = C + A = 151° 27' 40". 
.-. i(C + A)= 75° 43' 50". 



sin C 

Check: b 



\ ac sin B. 

_ a sin B 
sin J 



c = 52.925 log (c- a) = 1.43998 .-. i( c ~- 4 )= 54 ° 7'38" 

a= 25.384 colog (c + a) = 8.10619 l(C+A)= 75°43'50" 

C +a = 78^09 logtanKC+^)= P.5040Q adding , C=129 o 5r28 „ 

c-a = 27.541 log tan l(C-A) = 0.14077 subtracting, A= 21°36'12" 



log c = 1.72366 

log sin J3 = 9.67921 

colog sin C = 0.1 1484 

log 5 = 1.51771 
b= 32.939 



Check: log a = 1.40456 

log sin 5 = 9.67921 

colog sin A =0.43395 

log b = 1.51772 



log a = 1.40456 

log c = 1.72366 

log sin 5 = 9.67921 

log 2 A = 2.70743 

t= 509.83 = 254>965 



82 PLANE TRIGONOMETRY. 

Solve : (1) Given a = 0.325, c = 0.426, B = 48° 50' 10". 

(2) Given b = 4291, c = 3194, A = 73° 24' 50". 

(3) Given b = 5.38, c = 12.45, A = 62° 14' 40". 

4. Ambiguous cases. Since the required angle is found 
in terms of its sine, and since sin a = sin (180° — «), it fol- 
lows that there may be two values of a, one in the first, and 
the other in the second quadrant, their sum being 180°. In 
the following examples the student should note that all the 
marks of the ambiguous case are present. The solutions will 
show the treatment of the ambiguous triangle having no 
solution, one solution, two solutions. 

(a) Given 5 = 70, e = 40, C= 47° 32' 10". Solve. Why 
ambiguous ? 

. B = b sin O log b =1.84510 

c ' log sin 0= 9.86788 

colog c = 8.39794 

log sin^ = 0.11092 

.*. B is impossible, and there is no solution. Why? 

Show the same by sin C > -• 

J b 

(b) Given a = 1.5, c = 1.7, A = 61° 55 f 38". Solve. 

. n csinA log c= 0.23045 

a log sin A= 9.94564 

colog a = 9.82391 

log sin 0= 0.00000 

(7=90° 

and there is one solution. Why? Show the same by 

sin A = -. Solve for the remaining parts and check the 

c 
work. 



SOLUTION OF TRIANGLES. 83 

O) Given a = 0.235, b = 0.189, B = 36° 28' 20". Solve. 

A a sin B b sin C 

sin A = — - ? c = — : — —- 

6 sini* 

log a = 9. 37107 log b = 9. 27646 9. 27646 

log sin B=9. 77411 log sin (7=9. 99772 or 9. 28774 

colog b = 0.72354 colog sin B = 0. 22589 0. 22589 

log sin J. = 9.86872 log c = 9.50007 or 8.79009 

.4 = 47° 39' 25" c = 0.31628 or 0.06167 

or 132° 20' 35". 

.-. C=95°52' 15" or 11° 11' 5". 

Solve for A, and check. Show the same by sin B < — 

a 

Solve : 

(1) Given 5 = 216.4, c= 593.2, ^=98° 15'. 

(2) Given a = 22, b = 75, B = 32° 20'. 

(3) Given a = 0.353, c = 0.295, A = 46° 15' 20". 

(4) Given a = 293.445, b = 450, A = 40° 42'. 

(5) Given b = 531.03, <?= 629.20, ^=34° 28' 16". 



Solve completely, given : 










a 


b 


c 


A 




B 


C 


1. 50 


60 










78° 27' 47" 


2. 


10 


11 








93° 35' 


3. 4 


5 


6 










4. 




10 


109° 28' 


16" 


38° 56' 54" 




5. 40 


51 




49° 28' 


32" 






6. 352.25 


513.27 


482.68 










7. 0.573 


0.394 




112° 4' 








8. 107.087 






56° 15' 




48° 35' 




9. 




V2 


117° 




45° 




10. 197.63 


246.35 




34° 27' 








11. 4090 


3850 


3811 










12. 3795 










73° 15' 15" 


42° 18' 30" 


13. 
14. 


234.7 
26.234 


185.4 
22.6925 


84° 36' 




49° 8' 24" 




15. 273 


136 




72° 25' 13" 







84 PLANE TRIGONOMETRY. 

APPLICATIONS. 

70. Measurements of heights and distances often lead to 
the solution of oblique triangles. With this exception, the 
methods of Chapter V apply, as will be illustrated in the 
following problems. 

The bearing of a line is the angle it makes with a north 
and south line, as determined by the magnetic needle of the 
mariner's compass. If the bearing does not correspond to 
any of the points of the compass, it is usual to express it 
thus: N. 40° W., meaning that the line bears from N. 40° 
toward W. 

EXAMPLES. 

1. When the altitude of the sun is 48°, a pole standing on a slope 
inclined to the horizon at an angle of 15° casts a shadow directly down 
the slope 44.3 ft. How high is the pole? 

2. A tree standing on a mountain side rising at an angle of 18° 30' 
breaks 32 ft. from the foot. The top strikes down the slope of the moun- 
tain 28 ft. from the foot of the tree. Find the height of the tree. 

3. From one corner of a triangular lot the other corners are found to 
be 120 ft. E. by N., and 150 ft. S. by W. Find the area of the lot, and 
the length of the fence required to enclose it. 

4. A surveyor observed two inaccessible headlands, A and B. A was 
W. by N. and B, N.E. He went 20 miles N., when they were S.W. and 
S. by E. How far was A from B ? 

5. The bearings of two objects from a ship were N. by W. and N.E. 
by N. After sailing E. 11 miles, they were in the same line W.N.W. 
Find the distance between them. 

6. From the top and bottom of a vertical column the elevation angles 
of the summit of a tower 225 ft. high and standing on the same hori- 
zontal plane are 45° and 55°. Find the height of the column. 

7. An observer in a balloon 1 mile high observes the depression angle 
of an object on the ground to be 35° 20'. After ascending vertically and 
uniformly for 10 mins., he observes the depression angle of the same object 
to be 55° 40'. Find the rate of ascent of the balloon in miles per hour. 

8. A statue 10 ft. high standing on a column subtends, at a point 
100 ft. from the base of the column and in the same horizontal plane, the 
same angle as that subtended by a man 6 ft. high, standing at the foot 
of the column. Find the height of the column. 

9. From a balloon at an elevation of 4 miles the dip of the horizon 
is 2° 33' 40". Required the earth's radius. 



TRIANGLES — APPLICATIONS. 85 

10. Two ships sail from Boston, one S.E. 50 miles, the other N.E. by 
E. 60 miles. Find the bearing and distance of the second ship from the 
first. 

11. The sides of a valley are two parallel ridges sloping at an angle of 
30°. A man walks 200 yds. up one slope and observes the angle of eleva- 
tion of the other ridge to be 15°. Show that the height of the observed 
ridge is 273.2 yds. 

12. To determine the height of a mountain, a north and south base 
line 1000 yds. long is measured ; from one end of the base line the sum- 
mit bears E. 10° N., and is at an altitude of 13° 14'. From the other end 
it bears E. 46° 30' N. Find the height of the mountain. 

13. The shadow of a cloud at noon is cast on a spot 1600 ft. due west 
of an observer. At the same instant he finds that the cloud is at an ele- 
vation of 23° in a direction W. 14° S. Find the height of the cloud and 
the altitude of the sun. 

14. From the base of a mountain the elevation of its summit is 54° 20'. 
From a point 3000 ft. toward the summit up a plane rising at an angle 
•of 25° 30' the elevation angle is 68° 42'. Find the height of the mountain. 

15. From two observations on the same 

meridian, and 92° 14' apart, the zenith 

angles of the moon are observed to be 

44° 54' 21" and 48° 42' 57". Calling' the 

earth's radius 3956.2 miles, find the dis- 

, .-, [ £L ]/\Z- Zenith angle 

tance to the moon. 

16. The distances from a point to three 
objects are 1130, 1850, 1456, and the angles 

subtended by the distances between the three objects are respectively 
102° 10', 142°, and 115° 50'. Find the distances between the three objects. 

17. From a ship A running N.E. 6 mi. an hour direct to a port dis- 
tant 35 miles, another ship B is seen steering toward the same port, its 
bearing from A being E.S.E., and distance 12 miles. After keeping on 
their courses 1\ hrs., B is seen to bear from A due E. Find B's course 
and rate of sailing. 

18. From the mast of a ship 64 ft. high the light of a lighthouse is 
just visible when 30 miles distant. Find the height of the lighthouse, 
the earth's radius being 3956.2 miles. 

19. From a ship two lighthouses are observed due N.E. After sailing 
20 miles E. by S., the lighthouses bear N.N.W. and N. by E. Find the 
distance between the lighthouses. 

20. A lighthouse is seen N. 20° E. from a vessel sailing S. 25° E. A 
mile further on it appears due N. Determine its distance at the last 
observation. 




EXAMPLES FOR REVIEW. 

In connection with each, problem the student should review 
all principles involved. The following list of problems will then 
furnish a thorough review of the book. In solving equations, 
find all values of the unknown angle less than 360° that satisfy 
the equation. 

1. If tan a=-} t tan = J, show that tan (/? — 2 a) = T 2 T . 

2. Prove tan a + cot a = 2 esc 2 a. 

• A A A A 

3. From the identities sin 2 f- cos 2 — = 1, and 2 sin — cos— = sin A. 

9 9 9 9 



2 2 



prove 2 sin — = ± Vl + sin A ±Vl — sin A, 

A 



and 2 cos — = ± Vl + sin A =F Vl — sin A. 

4. Remove the ambiguous signs in Ex. 3 when A is in turn an angle 
of each quadrant. 

5. A wall 20 feet high bears S. 59° 5' E. ; find the width of its shadow 
on a horizontal plane when the sun is due S. and at an altitude of 60°. 

6. Solve sin x + sin 2 x + sin 3 x = 1 + cos x + cos 2 x. 

7. Prove tan- 1 - + tan" 1 - = -• 

2 3 4 

8. If A = 60°, 5 = 45°, C = 30°, evaluate 

tan A + tan B + tan C 



tan vl tan i? + tan B tan C + tan C tan .4 

9 Prove cos Q4 + B) cos C _ 1 — tan A tan Z? 
cos (A + C) cos B 1 — tan yl tan C 

10. Solve completely the triangle whose known parts are b = 2.35,, 
c = 1.96, C = 38° 45'.4. 

11. Find the functions of 18°, 36°, 54°, 72°. 

Let x = 18°. Then 2 x - 36°, 3 x = 54°, and 2 x + 3 x = 90°. 

12. If cot a = -, find the value of 

sin a + cos a + tan a + cot a + sec a + esc a. 
86 



EXAMPLES FOR REVIEW. 87 

13. Prove sin 3 <* sin 2 /? - sin 3 /? sin 2 <* =1 + ± cosa cos ^ 

sin 2 a sin /? — sin 2 /? sin a 

14. From a ship sailing due N., two lighthouses bear N.E. and 
N.N.E., respectively; after sailing 20 miles they are observed to bear 
due E. Find the distance between the lighthouses. 

15. Solve 1 — 2 sin x = sin 3 x. 



16. Prove sin-K.'— — = tan- 1 ^/-- 

V n J- h * h 



17. If cos 6 — sin = V2 sin 0, then cos + sin $ = V2 cos 0. 

18. Solve completely the triangle ABC, given a = 0.256, b = 0.387,. 
C = 102° 20'.5. 

19. Prove tan (30° + a) tan (30°- a) = 2cos2 ^~ 1 - 

2 cos 2 a + 1 

20. Solve tan (45° - 6) + tan (45° + 6) = 4. 

21. Prove sin 2 a cos 2 /? - cos 2 a sin 2 /? = sin 2 a — sin 2 /?. 

22. Prove cos 2 a cos 2 ft — sin 2 a sin 2 ft = cos 2 ft — sin 2 ft. 

23. A man standing due S. of a water tower 150 feet high finds its 
elevation to be 72° 30' ; he walks due W. to A street, where the elevation 
is 44° 50' ; proceeding in the same direction one block to B street, he finds 
the elevation to be 22° 30'. What is the length of the block between A 
and B streets ? 

24. Prove tan- 1 - + tan" 1 - + tan- 1 - + tan- 1 - = -• 

3 5 7 8 4 

25. If P = 60°, Q = 45°, R = 30°, evaluate 

sin P cos Q + tan P cos Q > 
sin P cos P + cot P cot R 

26. If cos (90° + ct) = -f, evaluate 3 cos 2 a + 4 sin 2 «. 

27. If sin B + sin C = m, cos 5 + cos C = n, show that tan — — — = — - 

2 n 

28. Show that sin 2 /? can never be greater than 2 sin /?. 

29. Prove sin -1 1 + sin -1 T 5 3 = tan -1 1|. 

30. Solve cot- 1 ^ + sin- 1 - VE = ~- 

5 4 

31. Solve sin -1 a: + sin -1 (l —x)= cos -1 a;. 

32. A man standing between two towers, 200 feet from the base of 
the higher, which is 90 feet high, observes their altitudes to be the same ; 
70 feet nearer the shorter tower he finds the altitude of one is twice that 
of the other. Find the height of the shorter tower, and his original 
distance from it. 



88 PLANE TRIGONOMETRY. 

33. Solve cos 3 (3 + 8 cos 3 (3 = 0. 

34. Solve cot m - tan (180° + m) = sec m 4- sec (90° - m). 

35. Solve 1 ~ tanf = 2cQS.2f. 

1 + tan t 

36. Prove cot ,4 + cot£= sin( ^ 4 + ^1 

sm J. sm 1? 

37. Prove cot P - cot Q = - sm ( P 7 Q) . 

sm P sin Q 

38. In the triangle ABC prove 

a = b sin C + c sin 5, 
6 = c sin A + a sin C, 
c = a sinE + & sin A. 

39. Solve completely the triangle, given 

a = 927.56, b = 648.25, c = 738.42. 

40. Prove cos 2 a - sin (30° + a) sin (30° - «)= |. 

.- -n , , cos 2 X — cos 4 X 

41. Prove tan 3 x tan x = — 

cos 2 x + cos 4 x 

42. Simplify cos (270° + a) + sin (180° + «) + cos (90° + a). 

43. Simplify tan (270° -6)- tan (90° + 6) + tan (270° + 0). 

44. Solve cos 3 cf> — cos 2 <£ + cos cj> = 0. 

45. Solve cos J. + cos 3 A + cos b A + cos 7 ^4 = 0. 

46. The topmast of a yacht from a point on the deck subtends the 
same angle a, that the part below it does. Show that if the topmast be 
a feet high, the length of the part below it is a cos 2 a. 

47. A horizontal line AB is measured 400 yards long. From a point 
in AB a balloon ascends vertically till its elevation angles at A and B 
are 64° 15' and 48° 20', respectively. Find the height of the balloon. 

48. If cos (f> = n sin a, and cot <j> = , prove cos (3 = . 

tan/2 VI + n 2 cos 2 a 

49. Find cos 3 a, when tan 2 a = — f . 

50. Solve completely the triangle, given a = 0.296, B - 2S° 47'.3, 
C = 84° 25'. 

51. Evaluate sin 300° + cos 240° + tan 225°. 

52. Evaluate sec ^ - esc ^ + tan i?. 

o o o 



EXAMPLES FOR REVIEW. 89 

53. If ^^shittcosy-sinffsiny 

cos a cos y — cos (3 sin y 
and tan <£ = Bin « sin y - sin /3 cos Y > 

cos a sin y — cos p cos y 

show that tan(0 + <f>) = tan(a + /?). 

54. If tan 466° 15' 38" = - ^, find the sine and cosine of 233° 7' 49". 

ec T3 esc a — cot a sec a — tan a 

55. Prove = 

sec a + tan a esc a + cot a 

56. Prove «*(« - 3ff)- cos(3 « - ff) = 2 gin(a _ o )# 

sin 2 « + sin 2 y3 

57. Prove sin 80° = sin 40° + sin 20°. 

58. Prove cos 20° = cos 40° + cos 80°. 

59. Prove 4 tan -1 tan" 

5 239 4 

60. From the deck of a ship a rock bears N.N.W. After the ship 
lias sailed 10 miles E.N.E., the rock bears due W. Find its distance 
from the ship at each observation. 

61. Find the length of an arc of 80° in a circle of 4 feet radius. 

62. Given tan = f , tan <£ = T % evaluate sin(0 + <f>) + cos(# — <£). 

63. If tan = 2 tan <j>, show that sin (6 + </>) = 3 sin(0 - <f>). 

64. Prove cos(a + ft)cos («-ff) + sin(tt + ff) sin (<*-/?) = * ~ tan ^ - 

1 -j- tan - £> 

65. Solve 4 cos 2 + 3 cos = 1. 

66. Solve 3 sin a = 2 sin (60° - <*)• 

67. Prove (sin a - esc a) 2 - (tan « - cot a) 2 + (cos « - sec a) 2 = 1. 

68. Prove 2 (sin 6 a + cos 6 a) + 1 = 3 (sin 4 os + cos 4 a). 

69. Prove esc 2 /J + cot 4 = cot (3 - esc 4 /?. 

70. If tan » = — , cos 2 o = — , then esc ^-—2 = 5VI3. 

F 12 H 625 2 

71. Solve completely the triangle, given 

a = 0.0654, b = 0.092, £ = 38°40'.4. 

72. Solve completely the triangle, given 

6 = 10, c = 26, B = 2'2° 37'. 

73. A railway train is travelling along a curve of ^ mile radius at the 
Tate of 25 miles per hour. Through what angle (in circular measure) 
will it turn in half a minute ? 



90 PLANE TRIGONOMETRY. 

74. Express the following angles in circular measure : 

63°, 4° 30', 6° 12' 36". 

75. Express the following angles in sexagesimal measure : 

7T 3 7T 17 7T 

6 8 ' 64 ' 

76. If A, B, C are angles of a triangle, prove 

ABC 

cos A + cos B + cos C = 1 + 4 sin — sin — sin — 

77. Prove sin 2 x + sin 2 # + sin 2 2 = 4 sin x sin ?/ sin z, when #, 3/, 2 
are the angles of a triangle. 

78. Prove sec ct = 1 + tan a tan -• 

79. Prove sin 2 (a + /?) - sin 2 (a - /?) = sin 2 a sin 2 /?. 

80. Prove cos 2 (a + /?) - sin 2 (a - (3)= cos 2 a cos 2 /?. 

, xj sin 19 » + sin 17 » ft 

81. Prove 1— - — : £ = 2 cos 9 ». 

sin 10 j9 + sin Sp 

82. Consider with reference to their ambiguity the triangles whose 
known parts are : 

(a) a = 2743, b = 6452, B = 43° 15' ; 

(b) a = 0.3854, c = 0.2942, C = 38°20'; 

(c) &= 5, c = 53, £ = 15°22'; 

(d) a = 20, b = 90, ^ = 63° 28'.5. 

83. From a ship at sea a lighthouse is observed to bear S.E. After 
the ship sailed N.E. 6 miles the bearing of the lighthouse is S. 27° 30' E. 
Find the distance of the lighthouse at each time of observation. 

84. Prove sin < $ ± S *> + sin ( 3 i ± fl = 2 cos (0 + #. 

sin 2 + sin 2 <£ v ^ J 

85. Prove cos 15° - sin 15° = — • 

V2 

86. Show that cos (« 4- /?) cos (a - /?) = cos 2 a - sin 2 (3 

= cos 2 /? - sin 2 a. 

87. Show that tan (a + 45°) tan (a - 45°) = 2 sin ^ ~ * • 

2 cos 2 a - 1 

88. Solve sin (z + y) sin (x — y) = \, cos (a + y) cos (x — y) = 0» 

89. p r ove * + sin " = cos a = tan *. 

1 + sin « + cos a 2 



EXAMPLES FOR REVIEW. 91 

90. Prove tan 2 6 + sec 2 = cos { + sin \ ■ 

cos — sm 

91. If tan <f> =-, then a cos 2 <£ + 6 sin 2 <£ = a. 

a 

92. Prove sin" 1 ^ + cot" 1 3 = -• 

V5 4 

93. Solve cos A + cos 7 ^4 = cos 4 A. 

94. Two sides of a triangle, including an acute angle, are 5 and 7, 
the area is 14 ; find the other side. 

95. Show that 8co»8fl-2co8fl-cog5g = ^ 2 ft 

sm 5 ^ — 3 sm 3 + 4 sin 6 

96. A regular pyramid stands on a square base one side of which is 
173.6 feet. This side makes an angle of 67° with one edge. What is 
the height of the pyramid ? 

97. From points directly opposite on the banks of a river 500 yards 
wide the mast of a ship lying between them is observed to be at an eleva- 
tion of 10° 28'.4 and 12° 14'.5, respectively. Find the height of the mast. 

98. Show that (sin 60° - sin 45°) (cos 30° + cos 45°) = sin 2 30°. 

99. Find x if sin- 1 x + sin- 1 x = t- 

2 4 

100. Trace the changes in sign and value of sin a + cos a as a 
changes from 0° to 360°. 



CHAPTER VIII. 



MISCELLANEOUS PROPOSITIONS. 



71. The circle inscribed in a given triangle is often called 
the incircle of the triangle, its centre the incentre, and its 
radins is denoted by r. The incentre is the point of inter- 
section of the three bisectors of the angles of the triangle 
(geometry). 

The circle circumscribed about a triangle is called the 
circumcircle, its centre the circumcentre, and its radius R. 
The circumcentre is the point of intersection of perpendicu- 
lars erected at the middle points of the three sides of the 
triangle (geometry). 





Incircle. 



Circumcircle. Escribed circle opposite A. 

Fig. 37. 



The circle which touches any side of a triangle and the 
other two sides produced is called the escribed circle; its 
radius is denoted by r a , r b , or r c , according as the escribed 
circle is opposite angle A, B, or 0. 

Again, the altitudes from the vertices of a triangle meet 
in a point called the orthocentre of the triangle. 

Finally, the medians of a triangle meet in a point called 
the centroid, which is two-thirds of the length of the median 
from the vertex of the angle from which that median is 
drawn (geometry). 

Certain properties of the above will now be considered. 

92 



MISCELLANEOUS PROPOSITIONS. 



93 



72. To find the radius of the incircle. 

Let A, A', A, ;/ A"' represent 
the areas of triangles ABO, 
COB, A 00, BOA, respectively. 
Then 
A = A' + A" + A'" 
= i(fl + J"+c)r= sr. 




And since A = Vs(s - a)(s — 5)(s - c), (Art. 63) 

A V(s - a)(s - b)(s — ~g) 



r = 



Cor. To express the angles in terms of r and the sides, 
divide each member of the above equation by s — a. 



Then — — = J ( g ~ 6)0 -<0 = tan i ^. ( Art< 62 ) 



s — a 



In like manner tan \B = 

2 s- b 



tan 1(7 = 



— c 



To find the radius of the circumcircle. 





Fig. 39. 

In the figure ABC is the given triangle, and i'Ca diam- 
eter of the circumcircle. Then, angle A = A', or 180°— A! . 

.-. sin A = smA r . 

Since A' BO is a right angle, 

BO a 



sin A' = 



.-. B = 



A'C 2B 



sin A 



94 



Cor. 1. As above, 2 R = 



PLANE TRIGONOMETRY. 
b 



sin A sin B sin C 
another proof of the "law of sines." 

a 



, which is 



Cor. 2. From R = 
R = 



2 sin A 

abc 



, we have 



2 be sin A 4 A 
74. To find the radii of the escribed circles 



= - — , where A= area ABO. 




Represent areas ABC, BOA, 
AOC, BOO, by A, A', A", A'", 
respectively. Then r a is the 
altitude of each of the triangles 
BOA,AOC,BOC. 



Now 



A = A'-h A"- A" r 
= lr a c+lr a b-\r a a 
= ir a (c+b-ay=r a Q 
A 



ay 



In like manner, r b 



s — a 

A 



A 
— c 



75. The orthocentre. 

Denote the perpendiculars on the sides 
a, 6, c, by AP a , BP b , CP C , and let it be 
required to find the distances from their 
intersection to the sides of the tri- 
angle, and also to the vertices. 

OP b = AP b tan CAO. 

But AP b = c cos A, and OA = 90° -O. 




OP b = o cos A cot C = 

= 2 B cos A cos O. 



sin O 



cos A cos C. 

(Art. 73, Cor. 1) 



MISCELLANEOUS PROPOSITIONS. 95 

In like manner, 0P C = 2 B cos B cos A, 
OP a = 2BcosOcosB. 

Again, the distances from the orthocentre to the vertices 
are, 

n A AP h c cos A 





r ~~~cos6 7 .A0~ 


sin 




= 2 B cos A. 




Also, 


OB = 2B cos B, 




and 


00= 2 B cos a 






76. Centroid and medians. 

The lengths of the medians may be computed as follows : 

In the figure the medians to the 2? 

sides a, b, c, are AM a , BM b , CM C , 
meeting in the centroid 0. 

Now, by the law of cosines, from A t 
the triangle BM b C, 

BM b 2 = a 2 + M b C 2 - 2 a- M b C • cos 
= a 2 H ab cos O. 

j, , n a 2 + b 2 - c 2 

But, cos C = — , 

2 ab 

' . 7?^2_, /2 , & a 2 + b 2 -c 2 _ 2a 2 + 2c 2 -b 2 



whence, BM b = J V2 a 2 + 2 c 2 - b 2 =. ±^a 2 + c 2 + 2 ac cos B 

a? + <?-b 2 j> 

since = cos B. 

2 ac 

In like manner, 



CM C = \ V2 b 2 + 2 a 2 - c 2 = i V6 2 + a 2 + 2 ba cos (7, 



and AM a = i V2 c 2 + 2 6 2 - a 2 = J Vc 2 + 6 2 + 2 c6 cos J.. 



96 PLANE TRIGONOMETRY. 



EXAMPLES. 

1. In the triangle, a = 25, b = 35, c = 45, find B, r, r a . 

2. Given a = 0.354, b = 0.548, C = 28° 34' 20", find the distances to 
C and B, from the circumcentre, the ineentre, the centroid, and the 
orthocentre. 

3. In the ambiguous triangle show that the circum circles of the two 
triangles, when there are two solutions, are equal. 

4. Prove that i + i + i = l. 

r„. r h r„ r 



5. In any triangle prove A = V 



r r a r b r c . 

6. Prove that the product of the distances of the ineentre from the 
vertices of the triangle is 4 r 2 R. 

7. Prove that the area of all triangles of given perimeter that can be 
circumscribed about a given circle is constant. 

8. Prove that the area of the triangle ABC is itr(sin A + sin B-\- sin C). 



CHAPTER IX. 

SERIES — DE MOIVRES THEOREM — HYPERBOLIC 
FUNCTIONS. 

77. First consider some series by means of which loga- 
rithms of numbers and the natural functions of angles may 
be computed. For this purpose the following series is. 
important : 

* = 1+ ^ + I + S + "- 

It may be derived as follows : 
By the binomial theorem, 

1+1) 

nj 
., , 1 . nx(nx-V) 1 , nx(nx — Y)(nx—%) 1 , 

n \1 n L [o n 6 

x { x -l) x { x HX x -l 

= l+x + — + rg 

and if n increase without limit, 

= 1 + * + | + | + ... + | + ... 

This is called the exponential series, and is represented by 
e x , so that 

|_2_ [3_ \r_ 

It is shown in higher algebra that this equation holds for 
all values of x ; whence, if x = 1, 

97 



98 PLANE TRIGONOMETRY. 

This value of e is taken as the base of the natural or 
Naperian system of logarithms. 

This value e, however, is not the base of the system of logarithms 
computed by Napier, but its reciprocal instead. The natural logarithm 
is used in the theoretical treatment of logarithms, and, as will presently 
appear, it is customary to compute the common logarithm by first 
finding the natural, and then multiplying it by a constant multiplier 
called the modulus, Art. 82 ; i.e. in the Naperian system the modulus 
is taken as 1, and the base is computed. In the common system the 
base 10 is chosen and the modulus computed. 

78. From the exponential series the value of e may be 
computed to any required degree of accuracy. 



1+1+ l= 


= 2.5 


1 

15 


=0.1666666666 


1 

[4 


= 0.0416666666 


1 


: 0.0083333333 


1 

[6 


= 0.0013888888 


1 


= 0.0001984126 


1 


=0.0000248015 


1 

I? 


0.0000027557 


1 
[10 


0.0000002755 



Adding, e = 2.7182818, correct to 7 decimal places. 



SERIES. 99 

79. To expand a* in ascending powers of jr. 

Let a x = e z , then z = log, a x = x • log, a. (Arts. 35, 40) 

Substituting 

«--! + »• log, q + * 2(1 °g' a)2 + ggffi a)3 + •- 

Now put 1 + a for a, and 
(l + a)*=l + aj-l6g,(l + a) 

, ^pog e a + ^)i 2 , ^pog.a + ^)i 3 , ... 

[2 [3 

But by the binomial theorem, 

(1 + a y = 1 + za + -V— — ^ a 2 + -^ -^ ^ a 3 + • ••. 

If L£ 

Equating coefficients of # in the second members of the 
above equations, 

r>iu yyO /yx 

log e (l + a) = a- _ + _-_ + ...; 
or writing x for a, 

7^*j /y»0 /v>4 

z < /e (i + *) = *-| + |-| + -. 

In this form the series is of little practical use, since it 
converges very slowly, and only when x is between + 1 and 
— 1 (higher algebra). 

Put — x for x, and 

/y& sy*Q /y»4: 

log, (1 - x) = - x - - - - - ; 

.% log, (1 + x) - log, (1 - a;) 

, 1+X f . X* . X 5 . \ 

LofC 



100 



PLANE TRIGONOMETRY. 



Finally, put for #, and 

!og e ^t— = log, (n + V)- log e 7i 



1 + 1 



{2n + l 3\2n + lJ 5\2n + l 



V 



log 6 (w+l)=log e w+2^ „ * . + 1 



3, 1 



2ra+l S\2n+lJ 5V2n+ 
a series which is rapidly convergent. 



+ • 



T 



80. From this series a table of logarithms to the base e 
may be computed. 

To find log e 2 put n = 1. Then, since log e 1 = 0, the series 
becomes 



I 3 3 • 3 3 5 • 3 5 7 • 3 7 9 • 3 9 

+ — — + • — — + •••! =0.693147. 
^ 11. a 11 ^ 13- 3 13 J 

The computations may be arranged thus : 






3 


2.00000000 






9 


.mmmm 


= 


.6666Q661 


9 


.07407407- 


- 3 = 


.02469136 


9 


.00823045- 


- 5 = 


.00164609 


9 


.00091449- 


- 7 = 


.00013064 


9 


.00010161 - 


-9 = 


.00001129 


9 


.00001129- 


-11 = 


.00000103 




.00000125- 


-13 = 


.00000009 



.69314717 
whence log e 2 = 0.693147, correct to 6 decimal places. 
To find log e 3, put n — 2, and 



log. 8 -log. 2 + 2(1 + ^4 



1 + JL + A; + 



•)• 





SERIES. 




5 


2.00000000 




25 


.40000000 


.40000000 


25 


.01600000-3 = 


.00533333 


25 


.00640000-5 = 


.00012800 


25 


.00025600-7 = 


.00000366 




.00000102-9 = 


.00000011 




.40546510 




log. 2 = 


.69314717 



101 



... log, 3 = 1.098612, 
correct to 6 decimal places. 

log, 4 = 2 x log e 2, log, 6 = log e 3 + log e 2, etc. (Why ?) 
The logarithms of prime numbers may be computed as above 
by giving proper values to n. 

81. Having computed the logarithms of numbers to base e, 
the logarithms to any other base may be computed by means 
of the following relation : 



Let 


log« n = x ; 


then a x = n. 






Also, 


log, ri = y ; 

.-. a x 


then 5 y = w, 
= 5*. 






Hence, 


log«(0 


= l0ga(^), 






and 


.'. X 


= y iog a 5. 






It follows that 


log« n 


= log 6 n ■ log a 6 ; 






whpnp.ft 


log & n 


— 1a"* <n ■ ■ 






\\ ±±\yXl\j\j 






This factor — 


— is called the modulus of the 


system 


of 



logarithms to base b. Using it as a multiplier, logarithms 
of numbers to base b are computed at once from the loga- 
rithms of the same numbers to any other base a. 



102 



PLANE TRIGONOMETRY. 



82. To compute the common logarithms. 

Common logarithms are computed from the Naperian by 



use of the modulus 



log, 10 



; i.e. 



log 10 n = log e n 



log, 10 
By Art. 80, log e 10 can be found, and 

= .434294, the modulus of the common system* 



log e 10 
Ex. Compute the common logarithms of: 

2, 3, 4, 6, 5, 10, 15, 216, 3375. 



COMPLEX NUMBERS. 

83. In algebra it is shown that the general expression for 
complex numbers is a -f- bi, where a represents all the real 
terms of the expression, b the coefficients of all the imagi- 
nary terms, and i is so defined that i 2 = — 1 ; whence 

i = V— 1, i 2 = — 1, i z = — i, z 4 = 1, etc. 

The laws of operation in algebra are found to apply to 
complex numbers. Moreover, it is further shown that if 
two complex numbers are equal, the real terms are equal, 
and the imaginary terms are equal; i.e. if 



then 



X- 



a-\-bi — c-\- di, 

a = b and c = d. 

Finally, the complex number 
may be graphically represented as 
follows : 



Y' 
Fig. 43. 



The real number is measured 
along OX, a units ; the imaginary 
parallel to 03 7 ", b units. The line 
r is a graphic representation of 
a + bi. 



DE MOIVRE'S THEOREM. 103 

Since a = r cos 6 and b = r sin 6, 

,\ a + bi = r (cos 6 4- i sin #). 

The properties of complex numbers are best developed by 
using this trigonometric form. If r be taken as unity, then 
cos 6 + i sin 6 represents any complex number. 

84. De Moivre's Theorem. To prove that, for any value 
of n, 

(cos 8 + i sin 0) n = cos nO + i sin w-0. 

I. When n is a positive integer. 
By multiplication, 

(cos a -f- i sin a) (cos j3 -{- i sin /3) 

= cos a cos /3 — sin a sin /3 + t (sin a cos /3 -f- cos a sin /3) 
= cos (a + £) + i sin (« + /3) . 
In like manner, 
(cos a + i sin a) (cos yS + z sin /3) (cos y + i sin 7) 

= cos(« -f y8 + 7) + * sin (« + /5 + 7) ; 
and finally, 

(cos «-H* sin a) (cos /3+i sin /3)(cos 7-M' sin 7) ••• to n factors 
= cos(« + /3 + 7 + -) + » sin(a + /3 + 7 + — )■ 
Now let a = /3 = 7 = •••, and the above becomes 
(cos a-\-i sin a) w = cos w« -f- i sin wa. 

II. When n is a negative integer. 
Let n = — m ; then 

(cos a + i sin «)" = (cos «4-« sin «) _m 

1 = 1 

(cos a + * sin a)*" cos ma + i sin ma 
cos ma — i sin ma 



(cos ma + i sin ma) (cos w?« — i sin ma) 
cos Ma — i sin 772a 



cos 2 77? « + sin 2 ma 
= cos ma — t sin ma = cos (— m)a + i sin (— m) a. 



104 PLANE TRIGONOMETRY. 

Substituting n for — m, the equation becomes 
(cos a + i sin a) n = cos na + i sin na. 

III. When w is a fraction, positive or negative. 

Let w = — , p and g being any integers. 

Now 

cos - + i sin - = cos a • - 4- i sin a - - = cos « + 1 sin a (by I). 
q qj q q 



Then 



a . . . a\ s . . N - 

cos - -f i sin - = (cos a.+ i sm a) q > 
q qJ 



Raising each member to the power p, 

p ( ' (*■ . . <*V p . . p 

(cos a + i sm « > = cos - + i sin - = cos — a + i sin — «. 



COMPUTATIONS OF NATURAL FUNCTIONS. 

85. The radian measure of an 
acute angle is greater than its sine 
and less than its tangent, i.e. 

sin a<a< tan a. 




N A 



Let a be the circular, or radian, 
measure of any acute angle A OP. 



Fig. 44. 

Then, in the figure, 

area of sector OAP < area of triangle OAT, 



i.e. i OA • arc AP <\OA- AT. 



Now, since 



But 



whence 



.-. nrcAP<AT. 

NP < arc AP, 

JSTP htcAP AT 
OP OP OP' 

■ circular measure of A OP = a 

sin a<a< tan a. 



NATURAL FUNCTIONS. 105 



86. Since sin a < a < tan a, 



sm a cos a 

Hence, however small a ma}' be, lies between 1 and 

_l sm a 

. "When a approaches 0, cos a approaches unity. 

cos a 

Therefore, by diminishing a sufficiently, we may make 

— differ from unity by an amount less than any assign- 
sin a 

able quantity. This we express by saying that when a 

approaches 0, approaches unity as a limit, i.e. — = 1, 

sin a sin a 

approximately. Multiplying by cos«(= 1, nearly), we have 

= 1, approximately. Whence, if a approaches 0, 

tan a 

tan « = sin a = «, approximately. 

87. Sine and cosine series. 

cos na -f i sin na = (cos a + i sin a) n , (De Moivre's Theorem). 

Expanding the second member by the binomial formula, 

it becomes, 

„ , „_i • • , n(n — 1) „_<> .o • o 
cos" a + n cos" l a • i sin a -\ ^— — ^ cos" - a . i A sir a 

\_ 

H ^ r^ - cos" 6 a . i i sim a 

LB 

. n(n — lVn — 2)(> — 3) „_ 4 . d • A 
+ — ^- — ^-cos" 4 a ■ z 4 sm 4 « + •••. 

Substituting the values of z 2 , **, i\ etc., we have 

, • • „ n(n — V) „_9 . 2 
cos ?i« + 8 sm na = cos" a 1 — — — -cos" - a sin- a 



n (n — 1) (n — 2) f ?i — 3) __. . 4 



[2 

cos" - * < 



I* 

+ z ( n cos" -1 a sin « — - v '" — — cos" _,J a sin 3 a + — )■ 



n (n - IX n - 2) _ oW _ 3 - 
I? 



106 PLANE TRIGONOMETRY. 



Equating the real and imaginary parts in the two members, 

„ n(n — 1) „ o . o 
cos na = cos n a ^— ^ cos n * a sin a 

,^0-1)0-2)0-3) n . .4 

i • «-i w(w-1)(m-2) w _q . o , 
and szw na — n cos n l a ^ ^ ^ cos ra * a szn"* a + • • •. 

L§ 

Ex. 1. Find cos 3ft; sin 3 ft. 

In the above put n = 3, and cos 3 ft = cos 3 ft — 3 cos ft sin 2 a 

= 4 cos 3 a — 3 cos «; 
also sin 3 a = 3 cos 2 ft sin ft — sin 3 ft 

= 3 sin ce — 4 sin 3 ft. 

2. Find sin 4 ft ; cos 4ft; sin 5 ft ; cos 5 ft. 

It will be noticed that in the series for cos na and sin na the terms 
are alternately positive and negative, and that the series continues till 
there is a zero factor in the numerator. 

88. If now in the above series we let na = #, then 

0fe_ 1 



/i « \ft 

COS U = COS" ft r^ COS n ~ z « Sill" 1 ft 



£ 



a\a J \ft / \« 

n 

6(0 -a) n _ 2 /sinft X2 

= COS 71 ft ^-: COS re ft 

[2 V ft 

^(^- a)( ^-2ft)(^-3ft) co ^_ 4 /sinftV ,„ 

|4 v « y 

If now remain constant, and ft decrease without limit, 

then will n become indefinitely great, and - and every 

ft 



NATURAL FUNCTIONS. 107 

power thereof, and cos a and every power of cos a will 
approach unity as a limit, so that 

^ , 02 04 06 

Similarly, s i„e = 9-| + | |_g + .... 

By algebra it is shown that these series are convergent for 

all values of 0. By their use we can compute values of sin 

and cos to any required degree of accuracy. 

# 3 2 h 

Show from the above that tan = 0-] 1 h •••• 

3 15 

Ex. I. Compute the value of sin 1°, correct to 5 places. 

make 6 the radian 



In 


[3 L5 II 


jasure of 


1O = 180 = " 0m5+ * 


Then, 


6 = 0.01745 + 




- = 0.0000008. 

\1 



.-. sin0 = 0.01745 +. 

The terms of the series after the first do not affect the fifth place, so 
that the value is given by the first term, an illustration of the fact that, 
if a is small, sin a = a, approximately. Compare the value of tan 1°. 

2. Show that sin 10° = 0.17365 ; cos 10° = 0.98481 ; sin 15° = 0.25882 ; 
cos 60° = 0.50000. 

3. Find the sine and cosine of 18° 30' ; 22° 15' ; 67° 45'. 

It is unnecessary to compute the functions beyond 30°, for since 

sin (30° + 0) + sin (30° - 0) = cos 6 (why?), 

.-. sin (30° + 0) = cos - sin (30° - 0) . 

So, also, cos (30° + 0) = cos (30° - 6) - sin 6. 

Giving proper values the functions of any angle from 30° to 45° are 
determined at once from the functions of angles less than 30°. 

Thus, sin 31° = cos 1° - sin 29° ; 

cos 31° = cos 29° - sin 1°. 

4. Find sine and cosine of 40° ; of 50°. 



108 PLANE TRIGONOMETRY. 

89. The following are sometimes useful in applied 
mathematics : 

Ex. 1. To find the sum of a series of sines of angles in A. P., such as 

sin a + sin (a + /?) + sin (a + 2 /?) + ••• + sin (a + [n - l]/3). 

2 sin a sin " = cos ( a — " ) — cos ( a + " ) , 
2 V 2^ V 27 

2 sin (a + y8)sin^ = cos (a + ^ ) - cos la + — ^), 
2 sin (a + 2/3) sin| = cosfa + ~^ - cos (a + -^Y 

2 sin (a + [n - 1]/?) sin^ = cos (a + 2n ~ 8 /?) - cos (a + 2 n ~ * /?) • 

Adding 
2 {sin a + sin (a + /3) + sin (a + 2 £) + ••• + sin (a + [w - 1] /?)} sin £ 

= cos^a ~ f ) - cos ( a + 2n 2 -1 /?) 
= 2sin(a + ^l^)sin|^ 
.-. sin a + sin (a + /?) + sin (cc + 2 /?)+••• + sin (a + [n - 1]/?) 
sin(«+^i/3)sin^ 

2 
Similarly it can be shown that 

cos a + cos (a + /?) + cos (a + 2 /?) + ••• + cos(« + [n - 1]/?) 

cos(« + ^i^sin^ 

= s^ ™ 

2 



HYPERBOLIC FUNCTIONS. 109 

90. The series e x = 1 + x -f- — + — 4- ••■ + — | — is proved 

in higher algebra to be true for all values of x, real or 
imaginary. Then if x = i0, 

- 1 -j| + 5-[« + -*^-i8 + iI-i7 + ->- 



» = cos + i sm (9 (Art. 87), 



In like manner, 


e '* = cos — i i 


Adding, 


a e ie + <r" 
cos 6 = — -!- 


subtracting, 


sm o = — -— 



are 



HYPERBOLIC FUNCTIONS. 

e ie — e~ ie e ie + e~ {i 
91. Since sin = — — , and cos = 

zi 2 

true for all values of 0, let = id. 

p~@ p" p® e~ 

Then, sin (z'0) = == i = i sinh 0, 

2 z 2 

and cos (i0) = — ^ = CO sh 0, 

, -, , , , ./jx sin (id) i sinh . , , n 

so that tan (iv) = -—- = — — = % tanh 0, 

cos (z0) cosh 6 

where sinh 6, cosh 0, tanh 0, are called the hyperbolic sine, 
cosine, and tangent of 0. The hyperbolic cotangent, secant, 
and cosecant of are obtained from the hyperbolic sine, 
cosine, and tangent, just as the corresponding circular func- 
tions, cotangent, secant, and cosecant, are obtained from 
tangent, cosine, and sine. The hyperbolic functions have 
the same geometric relations to the rectangular hyper- 



110 PLANE TRIGONOMETRY. 

bola that the circular functions have to the circle, hence the 
name hyperbolic functions. 

P e_ t> -e 2 

sinh 6 = 2 e , .-. csch 9 = j—^- 9 -> 

coshe^ v \ e , .-. sech9 = — -.; 

tanhQ = e9 ~ e ~ 9 , .: coth8 = e ° + e ~\ 



e u + e~ u e u - e~ u 

92. From the relations of Art. 91 it appears that to any 
relation between the circular functions there corresponds a 
relation between the hyperbolic functions. 

Since cos 2 (i0) 4- sin 2 (id) = 1, 

cosh 2 + i 2 sinh 2 = 1, 
or cosh 2 — sinh 2 = 1. 

This may also be derived thus: 

cosh 2 - sinh 2 



/e e + g -* y 



= g 2» + 2 + g -2e _ g M + 2 _ g -2^ _ ^ 

4 
Also since 

sin (ia + z/3) = sin (ia) cos (z'/3) + cos (ia) sin (^/3), 
.•. i sinh (« + |8)= i sinh a cosh /3 + cosh a • i sinh /3, 
and sinh (a + j3)= sinh « cosh /3 + cosh a sinh ft. 

Let the student verify this relation from the exponential 
values of sinh and cosh. 

EXAMPLES. 
Prove 

1. cosh (« + /?)= cosh a cosh (3 + sinh a sinh j3. 

2. cosh (a+ (3)- cosh (a — ft) =2 sinh a sinh /?. 

3. cosh 2 0=1 + 2 sinh 2 0=2 cosh 2 - 1. 

4. sinh 2a = 2 sinh ot cosh a. 



EXAMPLES. HI 



5. co8hg =A / 1+ ^ 8hg ; sinh^V^f- 1 . 

6. sinh 3 = 3 sinh + 4 sinh 3 6. 

7. sinh 6 + sinh <£ = 2 sinh ^_£ cosh ^-=-£ • 

8. sinh a + sinh (<* + /?)+ sinn ( a + 2 P) + •"+ sinn (« + [ n _ 1]/*) 

sinh f « + n ~ (3 J sinh^/2 

sinh^ 
2 

9. tanh(fl+<fr) = tanh 0+ tanh ^ 

V ^ 1 + tanh tanh </> 



10. sinh -1 x — cosh -1 Vl + x' 2 = tanlr 



VI + x* 

11. cosh (a + /?) cosh (a — (3) =cosh 2 a + sinh 2 (3 = cosh 2 /? + sinh 2 a. 

12. 2 cosh n« cosh a = cosh (n + 1) a + cosh (« — 1) a. 

13. cosh a = I O + e-*) = 1 + — + — + .... 

|2 14 

14. sinh a = I (e a — e~ a ) = a -\ 1 h •••. 

y J [3 |_5 

15. tanh- 1 a + tanh- 1 6 = tanh- 1 — tA. 

1 + a& 



SPHERICAL TRIGONOMETRY. 

CHAPTER X. 

SPHERICAL TRIANGLES. 

93. Spherical trigonometry is concerned chiefly with the 
solution of spherical triangles. Its applications are for the 
most part in geodesy and astronomy. 

The following definitions and theorems of geometry are 
for convenience of reference stated here. 

A great circle is a plane section of a sphere passing 
through the centre. Other plane sections are small circles. 

The shortest distance between two points on a sphere is 
measured on the arc of a great circle, less than 180°, which 
joins them. 

A spherical triangle is any portion of the surface of a 
sphere bounded by three arcs of great circles. We shall 
consider only triangles whose sides are arcs not greater than 
180° in length. 

The polar triangle of any spherical triangle is the triangle 
whose sides are drawn with the vertices of the first triangle 
as poles. If ABO is the polar of A'B'C, then A'B'O' is the 
polar of ABO. 

In any spherical triangle, 

The sum of two sides > the third side. 

The greatest side is opposite the greatest angle, and conversely. 
Each angle < 180° ; the sum of the angles > 180°, and 
< 540°. 

Each side < 180° ; the sum of the sides < 360°. 

112 



SPHERICAL TRIANGLES. 113 

The sides of a spherical tria?igle are the supplements of the 
angles opposite in the polar triangle, and conversely. 

If two angles are equal the sides opposite are equal, and 
conversely. 

The sides of a spherical triangle subtend angles at the 
centre of the sphere which contain the same number of angle 
degrees as the arc does of arc degrees ; i.e. an angle at the 
centre and its arc have the same measure numerically. 

The arc does not measure the angle for they have not the same unit 
of measurement, but we say they have the same numerical measure ; i.e. 
the arc contains the unit arc as many times as the angle contains the 
unit angle. 

The angles of a spherical triangle are said to be measured 
by the plane angle included by tangents to the sides of the 
angle at their intersection. They have therefore the same 
numerical measure as the dihe- 
dral angle between the planes B 
of the arcs. ^^"l\v 

In the figure the following ^^ \\ \\ 

^-"Mar / \ct N. 

have the same numerical meas- ° x^y— --_5_ I \ ^v 

ure : n. / Y^^~~~~~-^>c' 

arc a and angle a ; Ji\|| ^^^ 

arc b and angle j3; xl^^ 

A 
arc c and angle 7 ; Fig. 45. 

plane angle A'BC; 

spherical angle B and dihedral angle A-BO-C; 

spherical angle (7 and dihedral angle B-CO-A; 

spherical angle A and dihedral angle C-AO-B. 

A'C'B and C'A'B have not the same measure as spherical angles 
C and A, for BA', A'C, C'B are not perpendicular to OA or OC. 

94. In plane trigonometry the trigonometric functions 
were treated as functions of the angles. But since an angle 
and its subtending arc vary together and have the same 



114 



SPHERICAL TRIGONOMETRY. 



numerical measure, it is clear that the trigonometric ratios 
are functions of the arcs, and may be so considered. All 
the relations between the functions are the same whether we 
consider them with reference to the angle 
or the arc, so that all the identities of 
plane trigonometry are true for the func- 
tions of the arcs. 

Thus in the figure we may write, 

V • V 

sin a = - or sin a = - ; 
r r 

sin 2 a + cos 2 a = 1, or sin 2 a + cos 2 a = 1 ; 
cos 2 a = 2 cos 2 a — 1, or cos 2a = 2 cos 2 a — 1. 




GENERAL FORMULA FOR SPHERICAL TRIANGLES. 

95. The solutions of spherical triangles may be effected 
by formulae now to be developed: 

First it will be shown that in any spherical triangle 

cos a = cos b cos c + sin b sin c cos A 9 
cos b = cos c cos a + sin c sin a cos B, 
cos c = cos a cos 6 + sin a sin 6 cos C. 

The following cases must be considered : 
I. Both b and c < 90°. III. Both b and c> 90°. 

II. J > 90°, c < 90°. IV. Either 6orc= 90°. 

V. 6 = ^ = 90°. 

The figure applies to Case I. 

Let ABC be a spherical tri- 
angle, #, &, c its sides, and 
the centre of the sphere. 

Draw AC and AB' tangent 
to the sides 5, c at A. (The 
same result would be obtained 
by drawing AB\ AC perpen- 
dicular to OA at any point to 




GENERAL FORMULAE. 



115 



meet OB, OC.) Since these tangents lie in the planes of 
the circles to which they are drawn, they will meet OC and 
OB in C and B' , and the angle CAB' will be the measure 
of the angle A of the spherical triangle ABC. Since OAB\ 
OAC are right angles, A OB', AOC must be acute, and 
hence sides c, b are each < 90°. 
In the triangles CAB' and COB', 

CB' 2 = AC 2 + AB' 2 -2 AC ■ AB' cos CAB', 
and B' C 2 = OC 2 + OB' 2 - 2 OC • OB' cos C"0£'. 
Subtracting and noting that 

cos CAB' = cos A and cos COB' = cos a, 
we have 

= OC 2 - A C 2 + 6>^' 2 - ^L# /2 

+ 2 JO' • AB' cos .4 - 2 00" • 0£' cos a. 
But 6><7' 2 - J. C 2 = OA 2 and 0£ /2 - AB' 2 = OA 2 . 
Hence, = OA 2 + AC . AB' cos A - OC - OB' cos a ; 

eosa = dc'oW + oc'oW eosA ' 

.'. cos a = cos 6 cos c + sin 5 sin c cos A. 
Similarly, 

cos b = cos «• cos c -f- sin a sin c cos i?, 

and cos e = cos a cos b + sin a sin b cos O. 

These formulae are important, and should be carefully 

memorized. 

Q J? 

II. b>90°; c<90°. a^' 




In the triangle ABC, let b >90° 
and <?<90°. Complete the lune fig. 48. 

BACA'. Then in the triangle 
A'CB the sides a and J/O are both less than 90°, and by (I) 

cos A'B — cos A'C cos a + sin A'C sin a cos A'CB. 



116 SPHERICAL TRIGONOMETRY. 

But A'B = 180° -<?, A'C=180°-b, and ^'O5 = 180°- <7. 

.-. cos (180° - c) = cos (180° - 5) cos a 

+ sin (180° - b) sin a cos (180° - O) ; 

or — cos c = ( — cos &) cos a + sin 5 sin a ( — cos (7), 

and cos c = cos a cos 6 + sin a sin 6 cos Q. 

A similar proof will apply in case c > 90°, b < 90°. 

III. Both b and e> 90°. 



-a< 




In the triangle ABC, let both 
b and £ > 90°. Complete the 
lune ABA'O. Then since A'C 
and J/.B are both < 90°, 

cos a = cos A'G cos J/1? + sin A'C sin J/1? cos A! . 

But J' = A, A'Q= 180° - 5, A'B = 180° - <?. 

.-. cos a = cos (180° - 5) cos (180° - c) 

+ sin (180° - 5) sin (180° - <?) cos J ; 

or cos a = cos 5 cos c + sin 5 sin c cos J. 

Cases IY and Y are left to the student as exercises. 

96. Since the angles of the polar triangle are the supple- 
ments of the sides opposite in the 
first triangle, we have 

a' = 180° -A, b' = 180°-B, 
c < =180° -(7, A' = 180° -a. 

Substituting in 

cos a' = cos V cos c' 

+ sin V sin c' cos A! , 
we have 

- cos (180° - J) = cos (180° - B) cos (180° - (7) 

+ sin (180° - B) sin (180° - C) cos (180° - a); 

or —cos A= ( — cos !?)( — cos <7)-|-sin 1? sin (7( — cos a). 




GENERAL FORMULA. 117 

Changing signs, 

cos A = - cos B cos C + sin B sin C cos a. 

Similarly, cos B = — cos A cos C + sin ^1 sin C cos b, 

and cos C = - cos A cos B + sin ^1 sin 2? cos c. 

sin A sin J5 sin (7 



97. In any spherical triangle to prove 
Since cos A = 



sin a sin b sin c 
cos a — cos b cos <? 



sin b sin 



^ 2 



• 9 a 1 /cos a — cos b cos 

Sill 2 .A = 1 — ( : — - — ; 

V sin b sin c 



_ sin 2 b sin 2 c — (cos a — cos b cos c) 2 
sin 2 b sin 2 c 

_ (1 — cos 2 5) (1 — cos 2 c) — (cos a — cos b cos e) 2 
sin 2 b sin 2 c 



Hence, 



and 





A 
A 


1- 


cos 2 tf — 


cos 2 5 — cos 2 c + 2 


cos a cos 5 


cos c 


*J 






sin 2 5 sin 2 c 








sin 


Vl 


— cos 2 a 


— cos 2 6 — cos 2 c — 


2 


cos a cos 


b cos c 






sin 6 sin c 








sin 


VI 


— cos 2 a 


— cos 2 b — cos 2 <? — 


■2 


cos a cos 


6 cose 



sin a sin a sin b sin c 



By a similar process, — — - and — : will be found equal 

sin b sin c 



to the same expression. 



. sin A _ sinl? _ si n C 
sin a sin b sine 



118 SPHERICAL TRIGONOMETRY. 

98. Expressions for sine, cosine, and tangent of half an 
angle in terms of functions of the sides. 

A 

We have 2 sin 2 — = 1 — cos A 

A 

- 1 _ cos a ~~ cos ^ cos c 
sin b sin c 

_ cos b cos c + sin b sin c — cos a 
sin b sin c 

_ cos (b — c) — cos a^ 
sin b sin e 

Then 2 sin^ = 2 sin i ^ a + 1 7 e) si " * (a - b + c) (Art. 51) 

2 sin b sin <? 

_ 2 sin (s — 5) sin (s — c) 
sin 6 sin «? 

when 2s = #4-6-h<?. 



sin — = J s ln -(s-&)sin(s-c) . 
2 — ' sin 6 sin c 



Similarly, sin f = J sin^-c) S in(8 -«) 

^ 2 » sin •/. sin r. 



and sin g ' sin(s T &)sin(s-cE) . 

2 ^ sin a sin b 

Also from the relation 

2 cos 2 — = 1 + cos A 

A 

-i . cos a — cos b cos tf 
= 1 H . , . » 

sin o sin c 



we have cos 4 = >"* sin( ? - m) 

?< » sin /i sin ^ 



2 * sin 6 sin c 



Also, cosg=V Sin f Sin( .*~ &) 

2 > sin ^ sin •/, 



and C0S ^ = J singsm^-c) , 

2 * sin a sin b 



GENERAL FORMULA. 11$ 



From the above, . a 

sin 



A 2 ^ sm (s - b) sin (s - c) 

tan -s- = t - \— — i A7 — - — r— ^ 

^ * sin s sm (s - a) 



cos 2 



Also, telif = J 8ta(f-».)sln(s-c) t 

2 1 sm s sm (s — o) 

and tang =->BZ ^ is («-») . 

2 ' sin s sin (s - c) 

Compare the formulae thus far derived with the corresponding for- 
mulae for solving plane triangles. The similarity in forms will assist 
in memorizing the formulae for solving spherical triangles. 

99. From the formulae of Art. 96, the student can easily 
prove the following relations : 



Sm 2~^ siuUsinC 
where 2S = A + B + C. 



sing = ?. 



2 



a = J cosjS -B) cos (S-C) 
2 ^ sin B sin C 



cos| = ?, 



cos I 



tan ^ = J -cos.ScosCS-^l) 
2 *cos(S-JS)cosOS-C) 

tan| = ?, 



tan I 



120 SPHERICAL TRIGONOMETRY. 

100. Napier's Analogies. 



Since 





/sin (s — b) sin (s — c) 


tan 2 


^ sin s sin (s — a) 


tan — 


/sin (s — c) sin (s — a) 


2 


^ sin s sin (s — 6) 



-v 



sin 2 (s — 5) _ sin (s — 5) _ 
sin 2 (s — a) sin (s — a) 



by composition and division, 

tan h tan — 

2 2 sin (s — b) -f sin (s — a) 



tan^-tanf ™(«- &)-™(«-«0 

.A . B 

sin — sm — 

pAQ COS 

_2 2 _ sinj-(2*-a-6)cos£0-6) 

. A . B~ cos J (2 s — a — 6) sin %(a— b)' 

sin — — Sill , . _ : . 

2 2 (Art. 51) 

A B 

cos _ C0S - 

sin %(A + B) = tan i(2 s - a - 5) 
sin l(J. - B)~ tan i(a - 6) 

tan | 

since 2s — a — b = c. 



tan | {a — by 



1 sinl(4-J5) 
.-. tan | (a - V) = — ? tanf . 

sm\{A + B) Z 

To find an expression for tan ^ ( A — J5) we have only to 
consider the polar triangle, and by substituting 180° — A for 
a, etc., 180° — a for A, etc., we have the following relations : 

J (a - 5)= 1(180° -A- 180° + B) = —±(A - By 



GENERAL FORMULA. 121 

also, \(A-E)=- %(a -V)\ 

i-(A + B) = 1(180° - a + 180° - b)= 180°- i(a + 5); 

and |=90° -f 

The formula then becomes, applying Art. 29, 

. sin ~ (a - b) r 
tan 1 (.1 - B) = — s cot^. 

sin 1 (a + 6) 

Formulas for tan J (a + 5), tan |(^1 -f- B) are derived as 
follows : 

Since 

tan- • tan^ = J sin ( g -^) sin (>-<0 . J sin (s-e)sin(s-a) ^ 
2 2 ^ sin s • sin (s — a) ^ sin s • sin (s — 6) 



. A . B 
sin ^ sm 2 


sin (s — c) 


A B~ 

cos— cos— 

- — 


sins 



By composition and division, 

A B , . A . B 

cos— cos— + sm — sm — 

2 2 2 2 _ sin s -f- sin (s — c) 

A B . A . B sin s — sin (s — c^) 
cos— cos— — sm— sin— ° ° ^ ^ 

'A A A A 

wh ence <™ i ^ ~ ff = ^ * ffl + ft \ (Art. 51) 

oobK4+.^) tan* 

2 

since 2 s - e = a 4- i, 



or, 



cosk^-JB) _ 

tan I (« + &) = * tan^ 

cos\(A + B) l 



122 SPHERICAL TRIGONOMETRY. 

The value of tan J (A + B) is derived by substituting in 
terms of the corresponding elements of the polar triangle. 

cos \ (a - ft) - tan | (A + B) 

-cosj(a + -ft) G0t ^ 

-i cos -Aa — b) A c 
.-. tan*<^t + JB) = ^ icot^. 

cos I (a + ft) 

Similar relations among the other elements of the triangle 
may be derived, or they may be written from the above by 
proper changes of A, B, C, a, ft, c in the formulae. The stu- 
dent should write them out as exercises. 

101. Delambre's Analogies. 

Since sin J (A -f 2?) = sin — cos — + cos — sin — , 

Li Li L L 

then 



sin l(i + B) = sin (« - fr) + s in (g - a) . J sin g • sin (s - g) 
2 sin <? * sin a • sin ft 

(Art. 98) 



Hence, 



sin | (J, + 1?) _ sin (s - ft ) + sin (s - a) 
O sin c 



cos- 



2 sin - cos |- (a — ft) 



I sin - cos - 
2 2 



(Art. 51) 



cos \{a-b) „ 
and sin|(^ + B) = ^— cos|; 



In like manner derive 



sin \{a-b) „ 

sm\{A-B) = "— cos£; 

sin| 2 



RIGHT SPHERICAL TRIANGLES. 123 

cos|(a+6) „ 

em\{A + B) = sin^ 

cos ~ * 



sinl(a + 6) 

ms\(A-B) = — *— sin|. 



These formulse are often called Gauss's Formulae, but they were first 
discovered by Delambre in 1807. Afterwards Gauss, independently, dis- 
covered them, and published them in his Theoria Motus. 

102. Formulae for solving right spherical triangles are 
derived from the foregoing by putting (7 = 90°, whence 
sin (7=1, cos (7=0. 

cos c = cos a cos b + sin a sin b cos C (Art. 95) 

becomes cos c = cos a cos 6. (1) 

Substituting the value of cos a from (1), and simplifying, 







cos A = 

sin 


b 


sine 


(Art. 


95) 


becomes 




cos^=! an6 . 
tan c 








(2) 


Again, 




sin A sin 
sin a sin c 






(Art. 


97) 


in the right 


triangl 


e is 
smA = ^. 








(3) 



sine 

Dividing (3) by (2), 

sin a cos b sin a cos a cos b sin a 



tan ^L = 



cos c sin 6 cos c cos a sin 6 cos a sin 6 
since cos a cos 6 = cos c. 

.*. tan ^1 sin b = tan a. (4) 



124 SPHERICAL TRIGONOMETRY. 

From (4) tan a = tan A sin b, 

also, tan b = tan B sin a. 

Multiplying, tan a tan b = tan A tan B sin a sin 5, 
or, cot A cot 2* = cos a cos 6 = cos c. 

From (2) and (3), by division, 
tan 6 



(5) 



cos A tan c cos 



sm 



B sin 6 cos b 



= cos a. 



Sill c 

»•. cos ^1 = cos a sin U, 



(6) 



Let the student write formulae (2), (3), (4), (6) for B. 
It will be noticed that (1) and (5) give values for c only, 
while (2), (3), (4), (6) apply only to A and B. 



103. Formulae (l)-(6) are sufficient for the solution of 
right spherical triangles if any two parts besides the right 
angle are given. They are easily remembered by comparison 
with corresponding formulae in plane trigonometry. Two 
rules, invented by Napier, and called Napier's Rules of Cir- 
cular Parts, include all the formulae of Art. 102. 

Omitting C, and taking the comple- 
ments of A, c, and B, the parts of the 
triangle taken in order are a, b, 90° — A, 
90° - c, 90° - B. These are called 




the circular parts of the triangle. 

Any one of the five parts may be 

selected as the middle part, the two 

parts next to it are called the adjacent 

parts, and the remaining two the opposite parts. Thus, if 

a be taken as the middle part, 90° — B and b are the 

adjacent parts, and 90° — c, 90 — A the opposite parts. 



NAPIER'S RULES. 125 

Napier's Two Rules are as follows : 

The sine of the middle part equals the product of the tangents 
of the adjacent parts. 

The sine of the middle part equals the product of the cosines 
of the opposite parts. 

It will aid the memory somewhat to notice that i occurs in 
sine and middle, a in tangent and adjacent, and o in cosine 
and opposite, these words being associated in the rules. 

The value of the above rules is frequently questioned, 
most computers preferring to associate the formulae with 
the corresponding formulas of plane trigonometry. 

These rules may be proved by taking each of the parts as 
the middle part, and showing that the formulas derived from 
the rules reduce to one of the six formulas of Art. 102. 

Then, if b is the middle part, by the rules, 

sin b = tan a tan (90° — A) — tan a cot A, or tan A = — — -, 

sin b 

sin b = cos (90° - e) cos (90° - E) =sin c sin B, 

sin b 



or sin B 



sin c 



results which agree with (4). and (3), Art. 102. If any 
other part be taken as the middle part, the rules will be 
found to hold. 

104. Area of the spherical triangle. 

If r = radius of the sphere, 

E = spherical excess of the triangle =A + B+ (7 — 180°, 

A = area of spherical triangle, then by geometry 

A = Er* x -*-. 
180 

If the three angles are not known, E may be computed by 
one of the following methods, and A found as above. 



126 



SPHERICAL TRIGONOMETRY. 



Cagnoli's Method. 
sin^= an HA + B + C- 180°) 

c c 

= sin ^(A + B) sin— — cos ^(A + B) cos — 

1j 2 

. (7 
sm cos- 

= [cos \ (a - 5) - cos J (a + 5)] (Art. 101) 



cos- 



o ■ a . b 
2 sm - sm — 



2 V sin * sin (s — a) sin (s — 6) sin (s — c) 
sin a sin 6 



cos 



E __ V sin 8 sin (8 — a) sin Q - 6) sin (8 — c) 
2 2cos?cos^cos^ 



(Arts. 51, 98) 



2 2 2 

Lhuilier's Method. 

tan ^= *ml(A + B+ (7-180°) 
4 cosi(^l + ^H- (7-180°)' 

Now, multiply each term of the fraction by 
2cosl(A + £- (7+180°), 
and by Art. 51, (1) and (3), the equation becomes 



tan — = 
4 



C 
sin l ( A 4- i?) — cos— 

A 

cosl(J. + 1?)+ sin-^ 



[ 



cos \(a — 5) — cos- 



cos 



(7 



(Art. 101) 



[c 6 

cos J (a + 5) + cos - sin— 

_ sin \ (s — b) sin ^ (s — a) I sins sin(s — c ) 
s -,, \ ^sin(s — a)sin(* — 



COS - COS f (s — (?) 



A) 
(Art. 51) 



AREA OF SPHERICAL TRIANGLES. 127 

By Art. 52, introducing the coefficient under the radical, 



tan =^ = -v/tan! tan i(s -a)tani(s - 6)tanl(s - c), 



If two sides and the included angle are given, E may be 
determined as follows : 

cos^= cos I (A + B + C - 180°) 

Li 

c c 

= cos IQA + B) sin- + sin l (J. + B) cos-^ 

C 

= cos J (a + 6) sin 2 — 4- cos J (a — 6) cos 2 — (Art. 101) 



a b , • a . b ^ 

cos - cos - + sin - sm - cos C T 

2 2 2 2 



cos- 

9 



sin- sin- -2 sin- cos- 

But sin— = (Cagnoli's Method) 

cos| 



Dividing this equation by the above, 



. a . b . n 

sin - sin - sin C 

+ E 2 2 

tan— = 



2 a b , . a . b ^ 

cos -cos- + sin — sin - cos C 
9 9 9 9 



This formula is not suitable for logarithmic computations. 
Usually it is better to compute the angles by Napier's Analo- 
gies, and solve by A = Er 2 x — — • 

5 -> J lg0 



128 SPHERICAL TRIGONOMETRY. 



EXAMPLES. 

1. Show that cos a = cos b cos c + sin b sin c cos A becomes 

sec A = 1 + sec a, when a = b = c. 



2. If a + b + c = 7r, prove 

= tan — tan 

2 



(a) cos a = tan — tan — . 



/7 >, 9 ^4 cos a 

(o) cos 2 



2 sin b sin c 
(c) sin 2 — = cot b cot c. 
(r/) cos .4 + cos B + cos C = 1. 
(e) sin 2 — + sin 2 — + sin 2 *— = 1. 

■p (^ 

sin — cos \ (A — E) sin— sin \ (s — a) 

3. Prove = (Art. 104) 

. A a 

sin — cos - 

2 2 

4. Show that cos a sin b = sin a cos b cos C 4- sin c cos J. . 



CHAPTER XI. 

SOLUTION OF SPHERICAL TRIANGLES. 

105. According to the principles of spherical geometry 
any three parts are sufficient to determine a spherical tri- 
angle ; the other parts are computed, if any three are given, 
by the formulae of trigonometry. The known parts may be : 

I. Three sides, or three angles. 
II. Two sides and the included angle, or two angles and 
the included side. 

III. Two sides and an angle opposite one, or two angles 
and a side opposite one. 

It will appear that, as in plane geometry, III may be 
ambiguous. 

The signs of the functions in the formulae are important 
since the cosines and tangents of arcs and angles greater 
than 90° are negative ; whether the part sought is greater 
or less than 90° is therefore determined by the sign of the 
function in terms of which it is found unless this function 
be sine. In this case the result is ambiguous, since sin a 
and sin (180° — a) have the same sign and value. Thus if 
the solution gives log sin a = 9.56504, we may have either 
a = 21° 33', or 158° 27'. The conditions of the problem 
must determine which values apply to the triangle in 
question. 

The negative signs, when they occur, will be indicated 

thus : log cos 115° 20' = 9.63135", 

indicating, not that the logarithm is negative, but that 
in the final result account must be made of the fact that 
cos 115° 20' is negative. 

129 



130 SPHERICAL TRIGONOMETRY 

106. Formulce for the solution of triangles, 

j sin A _ sin B _ sin C 

sin a sin b sin c ' 



II. tan d = Jgjjjl^ftls in (8 - c ^ 

2 x sin 8 sin (« - a) 

III. tan g = V^g° s ^ cos ^ - 4H. 

2 * cos (S-B) cos (S-C) 

sin\(A-B) 

IV. ton | (a - 6) = f tan %. 

1 sini(^l + B) • 2 

2 

V. tan | (a + 6) = — f tan f . 

sin - (a — 6) ^ 

VI. tan | (^ - B) = — f cot £. 

2 sink«+6) 2 

2 

cos i (a - 6) r 
VII. ta n |U. + *) = _*— cotf. 

VIII. A = 2.,^, 

where E is determined by 

tan j=^tan|tan*(s-«)tan*(s-&)tani(s-c). 

Right triangles may be solved as special cases of oblique 
triangles, or by the following : 

(1) cos c = cos a cos b . (4 ) tan A sin 6 = tan a. 

/^ „~o i tan b ,_, 

(^; cos ^=j7— • (5) cot A cot B = cos c. 

(3) sin ^=4^- (6) cos^ = cos«sin J B. 

v ■ sin c v y 

The formula to be used in any case may be determined by 
applying Napier's Rule of Circular Parts. 

107. In solving a triangle the student should select formulae 



MODEL SOLUTIONS. 131 

in which all parts save one are known, and solve for that 
one (see page 77). Referring to Arts. 105 and 106, it will 
appear that solutions are effected as follows : 

Case I by formulae II, or III, check by I. 
Case II by formulae VI, VII, I, or IV, V, I, check by IV 
or VI. 

Case III by formulae I, IV, or I, VI, check by VI or IV. 

MODEL SOLUTIONS. 

108. 1. Given a - 46° 24', b = 67° 14', c = 81° 12'. Solve. 



4 /sin (s — b) sin (s — c) , 5 



sin 





-b) 


sin 


0- 


c) 




sill s sin 


(* 


-«) 




(sin 


(« 


-a) 


sin 


o- 


ft) 



_ ..- , . tau | = fein( s - a) sin (,- c ) t 

Sill s sin (s — 6) 

sin a sin & 



2 * sin s sin (s — c) 



sin J. sin 5 



Arrange and solve as in Example 1, page 80. 

Am. A = 46° 13'.5, B = , C = 

Solve : (1) .1 = 96° 45', £ = 108° 30', C = 116° 15'. 
(Use formulae III in the same manner as in Example 1.) 

(2) a = 108° 14', b = 75° 29', c = 56° 37'. 

(3) A = 57° 50', B = 98° 20', C = 63° 40'. 

2. Given b = 113° 3', c = 82° 39', J. = 138° 50'. Solve. 

tan i (5 + C) = cos ^ 6 - c ) cot4 tan \ (B - C) = sin ^ % ~ c ] cot ^, 
2V y cos|(6 + c) 2 sinf(Hc) 2' 

1(5+ C)±H^-O=^or C, sina = sln ; 4s " lfe - 

sm 5 

C/^: tan^ = taD H & - g ) sin -K^+C). 
2 sin i (^ - C) 

6 = 113° 3' log cos i (b-c}= 9.98453 log sin | (&-c) = 9.41861 

c = 82° 39' colog eos | (b + c) = 0.86461" colog sin i (b + e) = 0.00409 

| (6 + c) = . 97° 51' 1q cQt £ = 9>57466 b t A = 9 57466 

i(6-c)= 15° 12' & 2 & 2 

1 A = 69° 25' l°g tan i ( 5 + c ) = 0-42380 " log tan i (5 - C) = 8.99736 
1(5+0 =110° 39' H5-C')=5°40'.6 

\(B-C)= 5°40',6 
.-. 5 = 116° 19'.6 
and C = 104°58'.4 



132 SPHERICAL TRIGONOMETRY. 

Check : 
log sin A = 9.81839 log tan * (b - c) = 9.43408 

log sin b = §.96387 log sin \ (B + C) = 9.97116 

cologs in 5 = 0.04756 colog sin i (£ - C) = 1.00474 

log sin a = 9.82982 log tan a = 0>40998 

a = 137° 29' 2 

a = 137° 29' 
Notice that tan h (B + C) is — . Hence, |(J5 + C)is greater than 90°, i.e. 110° 39'. 

Solve: (1) A= 68° 40', 5= 56° 20', c= 84° 30'. 
(Use formulae IV, V, I. Compare Example 2.) 

(2) a = 102° 22', b = 78° 17', C = 125° 28'. 

(3) A = 130° 5', B = 32° 26', c = 51° 6'. 

109. Ambiguous cases. By the principles of geometry the 
spherical triangle is not necessarily determined by two sides 
and an angle opposite, nor by two angles and a side opposite. 
The triangle may be ambiguous. By geometrical principles 
it is shown that the marks of the ambiguous spherical tri- 
angle are: 

1. The parts given are two angles and the side opposite 
one, or two sides and the angle opposite one. 

2. The side, or angle, opposite differs from 90° more than 
the other given side, or angle. 

3. Both sides, or angles, given are either greater than 

90°, or less than 90°. 
A In the right triangle ABC^ 

sin a = sin A sin c. (formula (3)) 

\ Therefore there will be no solution, one 
_3 solution, or two solutions, according as 
1 ' sin a = sin A sin c, i.e. according as «^ 

the perpendicular p. (See Art. 65.*) 
But the most expeditious means of determining the am- 
biguity is found in the solution of the triangle. The use 
of formula I gives the solution in terms of sine, so that it is 
to be expected that two values of the part sought may be 
possible ; and whether the triangle be ambiguous or not, 
there must be some means of determining which of the two 



AMBIGUOUS SPHERICAL TRIANGLES. 133 

angles, a and 180° — a, that have the same sine is to be used. 
If there are two solutions, both values are used. 

This is determined in the further solution of the triangle 
by formula VI, which may be written 

cos j- (A + O) tan j- (a + <?) 



tan - = 2-^ : — — — - — 

2 



cos i(A- 0) 



Now -<90°, whence tan — is +. Then if for both values 

of (7, found by the sine formula, the second member is +, 
there are two solutions ; if the second member is — for 
either value of (7, there is but one solution ; while if both 
values of make the second member — , there is no solution. 
The various cases will be illustrated by problems. 

3. Given a = 62° 15'.4, b = 103° 18'. 8, A = 53° 42'.6. Solve. 

sin B = sin6sin - 4 , tan - = cos ^ A + B > > tan ' ( a + h \ 

sin a 2 cos \ ( A — B) 

sin C = Slncsin - 4 - Check : cot £ = tan | (^ - £) sin i_(a + 5), 
sin a 2 sin \ (a — b) 

Solving the first formula gives 

log sin B = 9.94756, 
whence B x = 62° 24' .4, 

£ 2 = 117°35'.6. 

For each of the values B x and B. 2 , 

cos j- (A + B) tan % (a + b) 
cos \ (A — B) 

is + and therefore equal to tan — Hence there are two solutions. Find 

c = 153° 9'.6, or 70° 25'.4 
and C = 155° 43'.2, or 59° 6'.2 

4. Given a = 46° 45'.5, A = 73° 11'.3, B = 61° 18'.2. Solve. 

sin & = sin asinB } cot ^ = tan * ( A ~ B>} cos ?( a + b \ 

sin ^4 2 cos \ (a — &) 

sin c = sin a sin C . ace* : tan c - = *>™ I (<* -*>) *™l (A + B). 
sin A 2 sin $ (A - B) 



134 SPHERICAL TRIGONOMETRY. 



Solving for b gives 


log sin b = 9.82446, 


whence 


\= 41°52'.5, 


and 


6 2 = 138° 7'.5. 



For the value b l the fraction 

tan \ (A - B) cos | (a + b) 
cos \ (a — 6) 

is + , but for b 2 cos £ (a + 6) is — , making the fraction — , and hence it 

C 
can not equal cot — , which is + . There is then but one solution. Find 

C = 60° 42'.7, c = 41° 35M. 



5. Given a - 


= 162° 


30', 


.4 =49° 50', B = 57° 52'. 


Solve. 


Solving gives 






log sin b = 9.52274, 




whence 






b l= 19°27'.9, 
b 2 = 160°32'.l. 





For both values, \ and & 2 , cos \ {a + b) is — . Therefore, 

tan \ (A - B) cos \ (a + b) 

cos \ (a — b) 

C 
is — and not equal to cot — • Hence the triangle is impossible. 

Solve, testing for the number of solutions : 

(1) b = 106° 24'.5, c = 40° 20', C = 38° 45'.6. 

(2) a = 80° 50 , A = 131° 40', B = 65° 25'. 

(3) a = 60° 31 '.4, b = 147° 32'. 1, B = 143° 50'. 

(4) a = 55° 30', c = 139° 5', A = 43° 25'. 

RIGHT TRIANGLES. 

110. Right triangles are a special case of oblique triangles,, 
but are usually solved by formulse (1) to (6), Art. 106. 
Students should have no difficulty in applying these. 

Computers generally question the utility of Napier's Rules 
of Circular Parts. For those who prefer the rules a problem 
will be solved by their use. 



90-B 




SPECIES. 135 

6. Given c = 86° 51', B = 18° 3'.5, C = 90°. 

The parts sought are a, b, A, and it is immaterial which is computed 
first, a and A are adjacent to c and B, while b is the middle part of c 
and B. Then by Napier's first rule 

sin (90° - B) = tan (90° - c) tan a ; 

COS 5 T, , 

or tan a = = cos B tan c, 

cot c 

which is formula (2). 

By the same rule 90—^1 

sin (90° - c) = tan (90° - .4) tan (90° - B), 

or cot A = - — - = cos c tan B, formula (5). 

cot B v J 

Finally by the second rule 

sin b = cos (90° - c) cos (90° - B) = sin c sin B, formula (3). 
The solutions give a = 86° 41 '.2, b = 18° 1'.8, A = 88° 58'.4. Verify. 

111. Species. Two angles or sides of a spherical triangle 
are said to be of the same species if they are both less, or 
both greater, than 90°. They are of opposite species when one 
is greater and the other less than 90°. Since the sides and 
angles of a spherical triangle may, any or all, be less or 
greater than 90°, it is necessary in solutions to determine 
whether each part is more or less than 90°. The directions 
already given are sufficient in oblique triangles. In right 
triangles the sign of the function will determine if the solu- 
tion gives the result in terms of cosine or tangent, but not 
if the result is found in terms of sine. Thus in Example 6, 
above, we have log sin b — 9.49068, whence b = 18° 1'.8, or 

161°58'.2. By formula (4) sin b =1^^. Now sin b is 
J tan A 

always +, therefore, tan a and tan A must be of the same 
sign, whence in any right spherical triangle an oblique angle 
and its opposite side must be of the same species. 

Again by formula (1) cos e = cos a cos b. Now cos c is + 
or — according as c is less or greater than 90°. If then 
c<90°, cos a and cos b are of the same sign, but if c>90°, 
cos a and cos b are of opposite sign. Therefore, if the 



136 SPHERICAL TRIGONOMETRY. 

hypotenuse of a right spherical triangle is less than 90°, the 
other sides, and hence the angles opposite, are of the same 
species; but if the hypotenuse be greater than 90°, the other 
sides, and the angles opposite, are of opposite species. 

112. Ambiguous right triangles. 

When the parts given are a side adjacent to the right 
angle, and the angle opposite this side, the triangle is 
ambiguous, for solving for the hypot- 
A kj ) enuse by formula (3) gives 

AvXJ^ S • sin a 

V ^ 7^ sin c = - — -, 

jjv. ^y sin A 

c 
Fig. 54. from which there result two values of c. 

By the last rule of species it follows that 
to the values of c, one <90°, the other >90°, there will cor- 
respond two values for b, one of the same species as a, the 
other of opposite species. 

Clearly sin c ^ 1, according as sin a = sin A, and hence 
there will be no solution, one solution, or two solutions, 
according as sin a = sin A. 

Solve the spherical triangles, right angled at (7, given : 

(1) b = 73° 21'. 4, c= 84° 48'. 7. 

(2) c = 54°28', .£ = 128° 12'. 6. 

(3) b = 45° 42', B = 135° 42'. 

(4) a =108° 22'. 3, b = 120° 14'. 5. 

(5) a= 70° 50', A= 170° 40'. 

(6) b = 32° 8'. 4, B =46° 2'. 8. 

(7) b = 34° 28', c = 62° 50'. 

(8) c = 102° 35', .5 = 17° 45'. 

(9) a= 92° 16', £=57° 35'. 









EXAMPLES 




137 


Solve, given : 




EXAMPLES. 








a 


b 


c 


A 


B 


C 


1. 


97° 35' 


27° 8'.4 


119° 8'.4 








2. 




67° 33'.4 


94° 5' 


99°57'.6 






3. 


40° 20' 


70° 40' 




40° 






4. 




82°39'.5 




116° 20' 




70° 7 


5. 


155° 47'.1 




110°46'.4 






90° 


6. 








49°44'.3 


121°10'.4 


26° 6'.3 


7. 


144° 10' 




41°44'.2 




130° 




8. 




127° 30' 




132° 16' 


139° 44' 




9. 




155° 5'.3 




110° 10' 




70°20'.8 


10. 


62° 42' 






50° 12' 


58° 8' 




11. 


120° 30' 


70°20'.3 


69° 35' 








12. 
13. 
14. 


50° 15' 
84°14'.5 






75° 30' 
116° 20' 


104° 59'.1 

32°26'.l 

« 


90° 

138° 50'.2 
36°45'.4 


15. 


100° 


50° 


60° 








16. 






87° 12' 


88° 12' 




90° 


17. 


63° 50' 


80° 19' 




50° 30' 






18. 








34° 15' 


42°15'.2 


121°36'.2 


19. 


50° 






63° 15' 




90° 


20. 


159° 50' 






159° 43' 


123° 40' 




21. 


124°12'.5 


54° 18' 


97°12'.5 








22. 








48°31'.3 


62°55'.7 


125° 18'.9 


23. 


76° 36' 




40° 20' 




42° 15'.2 




24. 






28°45'.l 


44°22'.2 


122°25'.l 




25. 




44° 53' 


53° 52' 






90° 



26. 98°21'.7 109°50'.4 115°13'.5 

27. 99°40'.8 64°23'.2 95°38M 



138 SPHERICAL TRIGONOMETRY. 

APPLICATIONS TO GEODESY AND ASTRONOMY. 

113. Geodesy is concerned in measuring portions of the 
earth's surface, considering the earth as a sphere. 

To find the distance on the earth's surface between two 
points whose latitudes and longitudes 
-\ are known. 

\ If A and B are two places on the 

\ , earth, P the north pole, UCDU' the 
^^A equator, and PEP' the principal merid- 
J ian, e.g. the meridian of Greenwich, 
__ ^/ and if the latitude and longitude of A 

„ p and B are known, then AB can be 

Fig. 55. 

computed. 

For AP = 90° - latitude A, 

BP = 90° - latitude B, 

angle APB = longitude A — longitude B. 

*\ two sides and the included angle of the triangle APB are 
known, and AB can be computed. 

Ex. 1. Find the distance between Ann Arbor, 42° 19' N., 83° 43'. 8 W., 
and San Juan, 18° 29' N., 66° 7' W. 

2. How fur is Manila, 14° 36' N., 120° 58' E., from Honolulu, 
21° 18' N., 157° 55' W.? Honolulu from San Francisco, 37° 47'. 9 N., 
122° 24'.5 W. ? San Francisco from Manila ? 

114. The celestial sphere. The heavenly bodies appear to 
be situated on a sphere of indefinitely great radius with the 
centre at the point of observation. This is called the celes- 
tial sphere. 

A tangent plane to the earth at the point of observation 
cuts the celestial sphere in a great circle called the horizon. 

The points of the horizon directly south, west, north, east 
are called the south, west, north, east points. 

A vertical line through the point of observation cuts the 
celestial sphere above in the zenith, and below in the nadir, 
the zenith and nadir being poles of the horizon. 



APPLICATIONS. 139 

The earth's axis produced is the axis of the celestial sphere, 
cutting it in the north and south poles of the equator. 

The altitude of a star is its distance from the horizon 
measured on an arc of a great circle drawn through the star 
and the zenith. 

The azimuth, or bearing, of a star, is the arc of the horizon 
measured from some fixed point to the foot of the great 
circle through the star and the zenith. The fixed point is 
usually the south point. 

The declination of a star is its distance from the celestial 
equator. The circle drawn through the pole and the star is 
the hour circle, and the angle at the pole between the prime 
meridian and the hour circle is the hour angle of the star. 

Let an observer be at on the surface 
of the earth, and let P be the position s^TT^^n 

oi a star. /\A ' >' \ 

Then Z is the zenith, Z' the nadir, H i^p^^i).---^-] H ' 
PJQU' the celestial equator, JN" its north \ \v^^2v ' 
pole, S its south pole, HRH' the horizon, s^J__^^ 

NPS the meridian, or hour circle, of P, z 

and ZNP the hour angle. The declina- 
tion of the star is PQ, its altitude PR, and its azimuth, or 
bearing, NZP. The astronomical triangle NZP can be 
solved if any three of its parts are known. 

EXAMPLES. 

1. What will be the altitude of the sun at 9 a.m. in Detroit, 
lat. 42° 20' N., its declination being 27°30'.5? 

2. At what time will the sun rise at San Francisco, lat. 37° 47'.9, if 
its declination is 32° 46'.2 ? 

3. Find the azimuth and altitude of a star to an observer in 
lat. 42° 20' X., when the hour angle of the star is 3 h. 42.3 m. E., and 
the declination is 42° 31' BT. 

4. The latitude of Sayre Observatory is 40° 36 r .4 N. ; the sun's alti- 
tude is 47° 15'.3, its azimuth 80° 23M. Find its declination and hour 
angle. 

5. At Ann Arbor, March 13, 1891, the altitude of Kegulus is 32° 10'.3, 
and the azimuth is 283° 5'.1. Find the declination and hour angle. 



FIVE-PLACE 
LOGARITHMIC AND TRIGONOMETRIC 

TABLES 



ADAPTED FROM GAUSS'S TABLES 

BY 

ELMER A. LYMAN 

MICHIGAN STATE NORMAL COLLEGE 
AND 

EDWIN C. GODDARD 

UNIVERSITY OF MICHIGAN 



>J*Kc 



ALLYN AND BACON 
Boston anti Chicago 



COPYRIGHT, 189 9, BY 
ELMER A. LYMAN and 
EDWIN C. GODDARD. 



Norfoooti ^rrss 

J. S. Cushing & Co. — Berwick & Smith 
Norwood Mass. U.S.A. 






4~ 



TABLE I. 

THE COMMON LOGARITHMS OF NUMBERS 
FROM 1 TO 10009. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 8 9 


P.P. 


100 


00 000 


043 


087 


130 


173 


217 


260 


303 346 389 




IOI 


432 


475 


5i8 


56i 


604 


647 


689 


732 775 817 


44 43 42 


I02 


860 


903 


945 


988 


*03o 


*CV2 


*n5 *i57 *i99 *242 


1 


4/4 4/3 4/2 
8,8 8,6 8,4 
13,2 12,9 12,6 
17,6 17,2 16,8 
22,0 21,5 21,0 
26,4 25,8 25,2 
30,8 30,1 29,4 
35/2 34,4 33,6 
39/6 38,7 37,8 


IO3 


01 284 


326 


368 


410 


452 


494 


536 


578 620 662 


IO4 


703 


745 


787 


828 


870 


912 


953 


995 *036 ^078 


3 

4 
5 
6 


I05 


02 119 


160 


202 


243 


284 


325 


366 


407 449 490 


I06 


53i 


572 


612 


653 


694 


735 


776 


816 857 898 


IO7 


938 


979 


*oi9 


*o6o 


#100 


*i4i 


*i8i 


*222 *262 *302 


7 


I08 


°3 342 


383 


423 


463 


503 


543 


583 


623 663 703 


8 


IO9 


743 


782 


822 


862 


902 


941 


981 *o2i *o6o *IOO 


9 


110 


04139 


179 


218 


258 


297 


336 


376 


4i5 454 493 




in 


532 


57i 


610 


650 


689 


727 


766 


805 844 883 


41 40 39 


112 


922 


961 


999 


*038 


*077 


*n5 


*I54 


^192 ^231 ^269 






113 


05308 


346 


385 


423 


461 


500 


538 


576 614 652 


1 


4/i 4/° 3/9 
8,2 8,0 7,8 

12.3 12,0 11,7 

16.4 16,0 15,6 

20.5 20,0 19,5 

24.6 24,0 23,4 

28.7 28,0 27,3 

32.8 32,0 31,2 
36/9 36/O 35,1 


114 


690 


729 


767 


805 


843 


881 


918 


956 994 *032 


2 
3 
4 
5 
6 


us 


06 070 


108 


145 


183 


221 


258 


296 


333 37i 408 


116 


446 


483 


52i 


558 


595 


633 


670 


707 744 781 


117 


819 


856 


893 


930 


967 


*oc>4 ^041 


^078 #115 *i5i 


7 
8 


118 


07 188 


225 


262 


298 


335 


372 


408 


445 482 518 


119 


555 


591 


628 


664 


700 


737 


773 


809 846 882 


9 


120 


918 


954 


990 #027 


*o63 


*Q99 


*i35 


#171 3,207 *243 




121 


08 279 


314 


350 


386 


422 


458 


493 


529 565 600 


38 37 36 


122 


636 


672 


707 


743 


778 


814 


849 


884 920 955 


1 


3/8 3/7 3/6 
7/6 7/4 7/2 
ii,4 11, 1 10,8 
15,2 14,8 14,4 
19,0 18,5 18,0 


123 


99i 


*026 


*o6i 


55-096 


*I32 


*i67 


*202 


*237 #272 *307 


2 


124 


09342 


377 


412 


447 


482 


517 


552 


587 621 656 


3 
4 

5 


125 


691 


726 


760 


795 


830 


864 


899 


934 968 *oo 3 


126 


10 037 


072 


106 


140 


175 


209 


243 


278 312 346 


6 


22,8 22,2 21,6 


127 


380 


415 


449 


483 


517 


55i 


585 


619 653 687 


7 


26,6 25,9 25,2 


128 


721 


755 


789 


823 


857 


890 


924 


958 992 *025 


8 


30,4 29,6 28,8 


129 


11 059 


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126 


160 


193 


227 


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294 327 361 


9 


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130 


394 


428 


461 


494 


528 


56i 


594 


628 661 694 




131 


727 


760 


793 


826 


860 


893 


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959 992 *024 


35 34 33 


132 


12057 


090 


123 


156 


189 


222 


254 


287 320 352 






133 


385 


418 


45o 


483 


516 


548 


58i 


613 646 678 


1 

2 
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134 


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743 


775 


808 


840 


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937 969 *OOI 


135 


13033 


066 


098 


130 


162 


194 


226 


258 290 322 


136 


354 


386 


418 


45o 


481 


513 


545 


577 609 640 


137 


672 


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735 


767 


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333 


364 


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426 


457 


489 


520 551 582 


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32 31 30 


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15 229 


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564 


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2 
3 
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167 


197 


227 


256 


286 


316 


346 376 406 


146 


435 


465 


495 


524 


554 


584 


613 


643 673 702 


147 


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761 


791 


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938 967 997 


7 

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811 840 869 




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17 609 


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811 840 


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157 


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602 629 


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352 


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172 


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171 


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176 


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674 


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724 748 


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969 993 


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126 


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171 


194 


217 


240 


262 285 


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398 


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621 


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771 


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239 


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302 


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513 


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207 


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618 


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723 


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765 


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216 


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606 


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282 


3d 


321 


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361 


380 


400 


420 




221 


439 


459 


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518 


537 


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577 


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616 


19 


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635 


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733 


753 


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869 


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160 


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218 


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276 


295 


315 


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372 


392 


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411 


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468 


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324 


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231 


361 


380 


399 


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310 


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382 


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217 


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287 


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8 




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18 


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40 140 


157 


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209 


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278 295 


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312 


329 


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620 637 


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739 756 773 


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246 263 280 


296 313 


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259 


330 


347 


363 


380 


397 


414 430 447 


464 481 


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260 


497 


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547 


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581 597 614 


631 647 




261 


664 


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697 


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747 764 780 


797 814 


17 


262 


830 


847 


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913 929 946 


963 979 


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177 


193 


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226 


243 259 275 


292 308 


3 

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265 


325 


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357 


374 


390 


406 423 439 


455 472 


266 


488 


5°4 


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537 


553 


570 586 602 


619 635 


6 


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267 


651 


667 


684 


700 


716 


732 749 7 6 5 


781 797 


7 


11,9 


268 


813 


830 


846 


862 


878 


894 9ii 927 


943 959 


8 


x 3,6 


269 


975 


991 


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*io4 #120 


9 


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270 


43 136 


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169 


185 


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217 233 249 


265 281 




271 


297 


3 J 3 


329 


345 


361 


377 393 409 


425 44i 


16 


272 


457 


473 


489 


5o5 


521 


537 553 569 


584 600 


1 
2 
3 
4 
5 
6 


1,6 
3,2 
4,8 
6,4 
80 


273 


616 


632 


648 


664 


680 


696 712 727 


743 759 


274 


775 


791 


807 


823 


838 


854 870 886 


902 917 


275 


933 


949 


965 


981 


996 


*oi2 *028 *c>44 *c>59 *<>75 


276 


44091 


107 


122 


138 


154 


170 185 201 


217 232 


9,6 

11,2 


277 


248 


264 


279 


295 


311 


326 342 358 


373 389 


7 
8 


278 


404 


420 


436 


45i 


467 


483 498 514 


529 545 


12,8 


279 


560 


576 


592 


607 


623 


638 654 669 


685 700 


9 


HA 


280 


716 


731 


747 


762 


778 


793 809 824 


840 855 




281 


871 


886 


902 


917 


932 


948 963 979 


994 *OIO 


15 


282 


45 025 


040 


056 


071 


086 


102 117 133 


148 163 






283 


179 


194 


209 


225 


240 


255 271 286 


301 317 


1 


I ,5 


284 


332 


347 


362 


378 


393 


408 423 439 


454 469 


2 
3 
4 
5 
6 


3,° 
4,5 
6,0 
7,5 
9,o 
io,5 
12 


285 


484 


500 


5i5 


53o 


545 


561 576 591 


606 621 


286 


637 


652 


667 


682 


697 


712 728 743 


758 773 


287 


788 


803 


818 


834 


849 


864 879 894 


909 924 


7 
8 


288 


939 


954 


969 


984 


#000 


*oi5" *030 *045 


*o6o 3,075 


289 


46 090 


i°5 


120 


135 


'150 


165 180 195 


210 225 


9 


13/5 


290 


240 


255 


270 


285 


300 


3i5 33° 345 


359 374 




291 


389 


404 


419 


434 


449 


464 479 494 


509 523 


14 


292 


538 


553 


568 


583 


598 


613 627 642 


657 672 




293 


687 


702 


716 


73i 


746 


761 776 790 


805 820 


1 


A ,4 

2,8 

4,2 

5,6 
IP 
8,4 
9,8 

11,2 
12,6 


294 


835 


850 


864 


879 


894 


909 923 938 


953 967 


2 
3 
4 
5 
6 


2 9S 


982 


997 


*OI2 


*026 


*°4i 


*c>56 #070 *o85 


*ioo *ii4 


296 


47 129 


144 


159 


173 


188 


202 217 232 


246 261 


297 


276 


290 


305 


319 


334 


349 363 378 


392 407 


7 
8 

9 


298 


422 


436 


451 


465 


480 


494 5°9 524 


538 553 


299 


567 


582 


596 


611 


625 


640 654 669 


683 698 


300 


712 


727 


741 


756 


770 


784 799 813 


828 842 




.. 


L. 


1 


2 


3 


4 


5 6 7 


8 9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 8 


9 


P.P. 


300 


47712 


727 


741 


756 


770 


784 


799 


813 828 


842 




301 


857 


871 


885 


900 


914 


929 


943 


958 972 


986 




302 


48 001 


015 


029 


044 


058 


073 


087 


101 116 


130 




3°3 


144 


159 


173 


187 


202 


216 


230 


244 259 


273 


15 


304 


287 


302 


316 


330 


344 


359 


373 


387 401 


416 


30S 


430 


444 


458 


473 


487 


501 


515 


53o 544 


558 


2 


3,0 


306 


572 


586 


601 


6i5 


629 


643 


657 


671 686 


700 


3 


4,5 


3°7 


714 


728 


742 


756 


770 


785 


799 


813 827 


841 


4 


6,0 


308 


855 


869 


883 


897 


911 


926 


940 


954 968 


982 


5 


7,5 


309 


996 


*oio #024 


*Q38 ^052 


*o66 


*o8o 


3,094 *io8 


*I22 


6 

7 

8 


9/Q 

10,5 

12,0 


310 


49136 


150 


164 


178 


192 


206 


220 


234 248 


262 


311 


276 


290 


304 


3i8 


332 


346 


360 


374 388 


402 


9 


J 3,5 


312 


415 


429 


443 


457 


47i 


485 


499 


513 527 


541 




313 


554 


568 


582 


596 


610 


624 


638 


651 665 


679 




3i4 


693 


707 


721 


734 


748 


762 


776 


79o 803 


817 


14 


31S 


831 


845 


859 


872 


886 


900 


914 


927 941 


955 


316 


969 


982 


996 


*oio 


*024 


*Q37 *o5i 


#065 *079 


*092 


1 


i,4 

2,8 


3i7 


50 106 


120 


133 


147 


161 


174 


188 


202 215 


229 


2 


318 


243 


256 


270 


284 


,297 


3* 1 


325 


338 352 


365 


3 
4 
5 
6 


4^2 

5,6 
7,o 
8,4 


319 


379 


393 


406 


420 


433 


447 


461 


474 488 


501 


320 


515 


529 


542 


556 


569 


583 


596 


610 623 


637 


321 


651 


664 


678 


691 


705 


718 


732 


745 759 


772 


7 


9,8 


322 


786 


799 


813 


826 


840 


853 


866 


880 893 


907 


8 


11,2 


323 


920 


934 


947 


961 


974 


987 


*ooi 


*oi4 *028 


*Q4i 


9 


12,6 


324 


5io55 


068 


081 


095 


108 


121 


135 


148 162 


175 




3 2 S 


188 


202 


2i5 


228 


242 


255 


268 


282 295 


308 


326 


322 


335 


348 


362 


375 


388 


402 


415 428 


441 




327 


455 


468 


481 


495 


508 


521 


■534 


548 561 


574 


13 


328 


587 


601 


614 


627 


640 


654 


667 


680 693 


706 


329 


720 


733 


746 


759 


772 


786 


799 


812 825 


838 


1 
2 
3 


i,3 
2,6 
3,9 


330 


851 


865 


878 


891 


904 


917 


930 


943 957 


970 


33i 


983 


996 


*oo9 *022 #035 


*048 


*o6i 


*075 *o88 


*ioi 


4 


5,2 

6,5 
7,3 


332 


52 114 


127 


140 


153 


166 


179 


192 


205 218 


231 


5 
6 


333 


244 


257 


270 


284 


297 


310 


323 


336 349 


362 


334 


375 


388 


401 


414 


427 


440 


453 


466 479 


492 


7 
8 

9 


9, 1 
IO ,4 
n,7 


335 


504 


517 


530 


543 


556 


569 


582 


595 608 


621 


336 


634 


647 


660 


673 


686 


699 


711 


724 737 


75o 




337 


763 


776 


789 


802 


815 


827 


840 


853 866 


879 




338 


892 


905 


917 


93o 


943 


950 


969 


982 994 *oo7 




339 


53020 


033 


046 


058 


071 


084 


097 


no 122 


135 


to 


340 


148 


161 


173 


186 


199 


212 


224 


237 250 


263 


1 


1,2 


34i 


275 


288 


301 


3i4 


326 


339 


352 


364 377 


390 


2 


2,4 


342 


403 


4i5 


428 


441 


453 


466 


479 


491 504 


517 


3 


3,6 


343 


529 


542 


555 


567 


580 


593 


605 


618 631 


643 


4 


4,8 


344 


656 


668 


681 


694 


706 


719 


732 


744 757 


769 


5 
6 

7 


6,0 

7,2 

8,4 


345 


782 


794 


807 


820 


832 


845 


857 


870 882 


895 


346 


908 


920 


933 


945 


958 


970 


983 


995 *oo8 


*020 


8 


9,6 


347 


54 033 


045 


058 


070 


083 


095 


108 


120 133 


145 


9 


10,8 


348 


158 


170 


183 


195 


208 


220 


233 


245 258 


27O 




349 


283 


295 


307 


320 


332 


345 


357 


370 382 


394 




350 


407 


419 


432 


444 


456 


469 


481 


494 506 


5i8 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 9 


P.P. 


350 


54 407 


419 


432 


444 


456 


469 


481 


494 


506 518 




35i 


53i 


543 


555 


568 


580 


593 


605 


617 


630 642 




352 


654 


667 


679 


691 


7°4 


716 


728 


741 


753 765 




353 


777 


790 


802 


814 


827 


839 


851 


864 


876 888 




354 


900 


9i3 


925 


937 


949 


962 


974 


986 


998 #01 1 


1 
2 


10 

i,3 
2,6 


355 


55023 


o35 


047 


060 


072 


084 


096 


108 


121 133 


356 


145 


157 


169 


182 


194 


206 


218 


230 


242 255 


3 


3,9 


357 


267 


279 


291 


303 


315 


328 


340 


352 


3 6 4 376 


4 


5,2 


358 


388 


400 


413 


425 


437 


449 


461 


473 


485 497 


5 


6,5 


359 


509 


522 


534 


546 


558 


570 


582 


594 


606 618 


6 

7 
8 


7,8 
9,i 
IO ,4 


360 


630 


642 


654 


666 


678 


691 


703 


715 


727 739 


361 


75i 


763 


775 


787 


799 


811 


823 


835 


847 859 


9 


n,7 


362 


871 


883 


895 


907 


919 


93i 


943 


955 


967 979 




363 


991 


*oo3 


*oi5 3=027 


*Q38 


*o5o 


*o62 


*074 


*o86 #098 




364 


56 no 


122 


134 


146 


158 


170 


182 


194 


205 217 


12 


365 


229 


241 


253 


265 


277 


289 


301 


312 


324 33 6 


366 


348 


360 


372 


384 


396 


407 


419 


43i 


443 455 


1 


1 2 


367 


467 


478 


490 


502 


514 


526 


538 


549 


56i 573 




2^4 

3,6 
4,8 
6,0 
7, 2 


368 


585 


597 


608 


620 


632 


644 


656 


667 


679 691 


3 

4 

5 
6 


369 


703 


714 


726 


738 


750 


761 


773 


785 


797 808 


370 


820 


832 


844 


855 


867 


879 


891 


902 


914 926 


37i 


937 


949 


961 


972 


984 


996 


#008 


*oi9 *c>3 1 #043 


7 


8,4 


372 


57054 


066 


078 


089 


IOI 


113 


124 


136 


148 159 


8 


9,6 


373 


171 


183 


194 


206 


217 


229 


241 


252 


264 276 


9 


10.8 


374 


287 


299 


310 


322 


334 


345 


357 


368 


380 392 




375 


403 


415 


426 


438 


449 


461 


473 


484 


496 507 


376 


519 


530 


542 


553 


565 


576 


588 


600 


611 623 




377 


634 


646 


657 


669 


680 


692 


703 


7i5 


726 738 


11 


378 


749 


761 


772 


784 


795 


807 


818 


830 


841 852 


379 


864 


875 


887 


898 


910 


921 


933 


944 


955 967 


1 
2 
3 


i,i 
2,2 
3,3 


380 


978 


990 *OOI 


#013 *024 


*o35 *o47 ^058 *070 *o8i 


381 


58 092 


104 


115 


127 


138 


149 


161 


172 


184 195 


4 


4,4 


382 


206 


218 


229 


240 


252 


263 


274 


286 


297 3°9 


5 


5,5 


383 


320 


33i 


343 


354 


365 


377 


388 


399 


410 422 


6 


6,6 


384 


433 


444 


456 


467 


478 


490 


501 


512 


524 535 


7 
8 

9 


7,7 
8,8 
9,9 


385 


546 


557 


569 


580 


59i 


602 


614 


625 


636 647 


386 


659 


670 


681 


692 


704 


715 


726 


737 


749 760 




387 


771 


782 


794 


805 


816 


827 


838 


850 


861 872 




388 


883 


894 


906 


917 


928 


939 


95o 


961 


973 984 




389 


995 


*oo6 


*oi7 


*028 


*040 


*o5i 


#062 


*Q73 


*o84 *095 




390 


59 106 


118 


129 


140 


151 


162 


173 


184 


195 207 


1 


10 


39i 


218 


229 


240 


251 


262 


273 


284 


295 


306 318 


2 


2 ,° 


392 


329 


340 


35i 


362 


373 


384 


395 


406 


417 428 


3 
4 
5 
6 
7 


3,° 
4,o 
5,o 
6,0 
7,° 


393 


439 


450 


461 


472 


483 


494 


506 


517 


528 539 


394 


550 


56i 


572 


583 


594 


603 


616 


627 


638 649 


395 


660 


671 


682 


693 


704 


7i5 


726 


737 


748 759 


396 


770 


780 


791 


802 


813 


824 


835 


846 


857 868 


8 


8,0 


397 


879 


890 


901 


912 


923 


934 


945 


956 


966 977 


9 


Q,0 


398 


988 


■999 *OIO *02I 


*032 


*043 *o54 


*o65 *o 7 6 *o86 




399 


60 097 


108 


119 


130 


141 


152 


163 


173 


184 195 




400 


206 


217 


228 


239 


249 


260 


271 


282 


293 304 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


400 


60 206 


217 


228 


239 


249 


260 


271 


282 


293 


304 




401 


314 


325 


336 


347 


358 


369 


379 


390 


401 


412 




402 


423 


433 


444 


455 


466 


477 


487 


498 


509 


520 




403 


53i 


54i 


552 


563 


574 


584 


595 


606 


617 


627 




404 


638 


649 


660 


670 


681 


692 


703 


713 


724 


735 




405 


746 


756 


767 


778 


788 


799 


810 


821 


831 


842 


406 


853 


863 


874 


885 


895 


906 


917 


927 


938 


949 


11 


407 


959 


970 


981 


991 


#002 


#013 #023 #034 *o45-#°SS 






408 


61 066 


077 


087 


098 


109 


119 


130 


140 


151 


162 


1 


i y i 


409 


172 


183 


194 


204 


215 


225 


236 


247 


257 


268 


2 
3 
4 

5 
6 


2/2 

3/3 
4/4 
5/5 
6/6 


410 


278 


289 


300 


310 


321 


33i 


342 


352 


363 


374 


411 


384 


395 


405 


416 


426 


437 


448 


458 


469 


479 


412 


490 


5°o 


5ii 


521 


532 


542 


553 


563 


574 


584 


7 
8 


7,7 
8,8 


413 


595 


606 


616 


627 


637 


648 


658 


669 


679 


690 


414 


700 


711 


721 


73i 


742 


752 


763 


773 


784 


794 


9 


9,9 


415 


805 


815 


826 


836 


847 


857 


868 


878 


888 


899 




416 


909 


920 


930 


941 


95i 


962 


972 


982 


993 *oo3 




417 


62 014 


024 


034 


045 


055 


066 


076 


086 


097 


107 




418 


118 


128 


138 


149 


159 


170 


180 


190 


201 


211 




419 


221 


232 


242 


252 


263 


273 


284 


294 


304 


315 




420 


325 


335 


346 


356 


366 


377 


387 


397 


408 


418 


421 


428 


439 


449 


459 


469 


480 


490 


500 


5ii 


521 


10 


422 


531 


542 


552 


562 


572 


583 


593 


603 


613 


624 




423 


634 


644 


655 


665 


675 


685 


696 


706 


716 


726 


1 


J.,0 


424 


737 


747 


757 


767 


778 


788 


798 


808 


818 


829 


2 

3 
4 
5 
6 


2 /0 
3,0 

4/Q 
5/Q 
6,0 


425 


839 


849 


859 


870 


880 


890 


900 


910 


921 


931 


426 


941 


95i 


961 


972 


982 


992 *002 


*OI2 


*022 


*033 


427 


63043 


o53 


063 


073 


083 


094 


104 


II 4 


I24 


J 34 


7 
8 


7P 

8 


428 


144 


155 


165 


175 


i85 


195 


205 


215 


225 


236 


429 


246 


256 


266 


276 


286 


296 


306 


317 


327 


337 


9 


9/Q 


430 


347 


357 


367 


377 


387 


397 


407 


417 


428 


438 




43i 


448 


458 


468 


478 


488 


498 


508 


5l8 


528 


538 




43 2 


548 


558 


S68 


579 


589 


599 


609 


619 


629 


639 




433 


649 


659 


669 


679 


689 


699 


709 


719 


729 


739 




434 


749 


759 


769 


779 


789 


799 


809 


819 


829 


839 




435 


849 


859 


869 


879 


889 


899 


909 


919 


929 


939 


436 


949 


959 


969 


979 


988 


998 


*oo8 


*oi8 


*028 


*Q38 


a 


437 


64 048 


058 


068 


078 


088 


098 


108 


118 


128 


137 






438 


147 


157 


167 


177 


187 


197 


207 


217 


227 


237 


1 


°/9 


439 


246 


256 


266 


276 


286 


296 


306 


316 


326 


335 


2 

3 
4 
5 
6 


I /8 

2/7 

3/6 
4/5 


440 


345 


355 


365 


375 


385 


395 


404 


414 


424 


434 


441 


444 


454 


464 


473 


483 


493 


503 


513 


523 


532 


442 


542 


552 


562 


572 


582 


59i 


601 


611 


621 


631 


5/4 
6/3 


443 


640 


6^0 


660 


670 


680 


689 


699 


709 


719 


729 


7 
8 

9 


444 


738 


748 


758 


768 


777 


787 


797 


807 


816 


826 


7/2 

8,1 


445 


836 


846 


856 


865 


875 


885 


895 


904 


914 


924 




446 


933 


943 


953 


963 


972 


982 


992 ^002 


*on 


*02I 




447 


65031 


040 


050 


060 


070 


079 


089 


099 


108 


Il8 




448 


128 


137 


147 


157 


167 


176 


186 


196 


205 


215 




449 


225 


234 


244 


254 


263 


273 


283 


292 


302 


312 




450 


321 


33i 


34i 


35o 


360 


369 


379 


389 


398 


408 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 8 9 


P.P. 


450 


65321 


33i 


34i 


35o 


360 


369 


379 


389 398 408 




45i 


418 


427 


437 


447 


45 6 


466 


475 


485 495 504 




452 


5H 


523 


533 


543 


552 


562 


57i 


581 591 600 




453 


610 


619 


629 


639 


648 


658 


667 


677 686 696 




454 


706 


7i5 


725 


734 


744 


753 


763 


772 782 792 




455 


801 


811 


820 


830 


839 


849 


858 


868 877 887 


456 


896 


906 


916 


925 


935 


944 


954 


963 973 982 


10 


457 


992 *OOI 


*OII 


*020 


*030 


*°39 *c>49 


x.058 *o68 3,077 




458 


66087 


096 


106 


'lI5 


124 


134 


143 


153 162 172 


1 


±j\j 


459 


181 


191 


200 


2IO 


219 


229 


238 


247 257 266 


3 
4 
5 
6 


3P 

4,o 
5,o 
6,0 


460 


276 


285 


295 


3°4 


3H 


323 


332 


342 35i 36i 


461 


37o 


380 


389 


398 


408 


417 


427 


436 445 455 


462 


464 


474 


483 


492 


5° 2 


5ii 


521 


530 539 549 


7 
8 


7P 
8,0 


463 


558 


567 


577 


586 


596 


605 


614 


624 633 642 


464 


652 


661 


671 


680 


689 


699 


708 


717 727 736 


9 


9]° 


465 


745 


755 


764 


773 


783 


792 


801 


811 820 829 




466 


839 


848 


857 


867 


876 


885 


894 


904 913 922 




467 


932 


941 


95o 


960 


969 


978 


987 


997 *oo6 *oi5 




468 


67 025 


o34 


043 


052 


062 


071 


080 


089 099 108 




469 
470 


117 


127 


136 


145 


154 


164 


173 


182 191 201 




210 


219 


228 


237 


247 


256 


265 


274 284 293 


47i 


302 


311 


321 


330 


339 


348 


357 


367 376 385 


9 


472 


394 


403 


413 


4*2 


43i 


440 


449 


459 468 477 




473 


486 


495 


504 


514 


523 


532 


54i 


550 560 569 


1 


1,8 

2,7 

3,6 
4,5 
5,4 
6,3 

7,2 

8,i 


474 


578 


587 


596 


605 


614 


624 


633 


642 651 660 


2 
3 

4 

5 


475 


669 


679 


688 


697 


706 


715 


724 


733 742 752 


476 


761 


770 


779 


788 


797 


806 


8i5 


825 834 843 


477 


852 


861 


870 


879 


888 


897 


906 


916 925 934 


7 
8 


478 


943 


952 


961 


970 


979 


988 


997 


*oo6 *oi5 ^024 


479 


68034 


043 


052 


061 


070 


079 


088 


097 106 115 


9 


480 


124 


133 


142 


151 


160 


169 


178 


187 196 205 




481 


215 


224 


233 


242 


251 


260 


269 


278 287 296 




482 


3o5 


314 


323 


332 


34i 


35o 


359 


368 377 386 




483 


395 


404 


413 


422 


43i 


440 


449 


458 467 476 




484 


485 


494 


502 


5ii 


520 


529 


538 


547 556 565 




485 


574 


583 


592 


601 


610 


619 


628 


637 646 653 


486 


664 


673 


681 


690 


699 


708 


717 


726 735 744 


8 


487 


753 


762 


771 


780 


789 


797 


806 


815 824 833 


488 


842 


851 


860 


869 


878 


886 


895 


904 913 922 


1 


0,0 

1,6 
2,4 

3,2 


489 


93i 


940 


949 


958 


966 


975 


984 


993 *002 *OI I 


2 

3 
4 


490 


69 020 


028 


o37 


046 


055 


064 


073 


082 090 099 


491 


108 


117 


126 


135 


144 


152 


161 


170 179 188 


5 
6 


4,° 
4,8 
5,6 
6,4 
7,2 


492 


197 


205 


214 


223 


232 


241 


249 


258 267 276 


493 


283 


294 


302 


3ii 


320 


329 


338 


346 355 364 


7 
8 

9 


494 


373 


38i 


39° 


399 


408 


417 


425 


434 443 452 


495 


461 


469 


478 


487 


496 


504 


513 


522 531 539 




496 


548 


557 


566 


574 


583 


592 


601 


609 618 627 




497 


636 


644 


653 


662 


671 


679 


688 


697 705 714 




498 


723 


732 


74° 


749 


758 


767 


775 


784 793 801 




499 


810 


819 


827 


836 


845 


854 


862 


871 880 888 




500 


897 


906 


914 


923 


932 


940 


949 


958 9 6 6 975 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 8 9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 9 


P.P. 


500 


69897 


906 


914 


923 


932 


940 


949 


958 


966 975 




501 


984 


992 #001 


*oio 


*oi8 


*027 


*c>36 


*044 *o53 *o62 




502 


70 070 


079 


088 


096 


105 


114 


122 


131 


140 148 




5°3 


i57 


165 


174 


183 


191 


200 


209 


217 


226 234 




S04 


243 


252 


260 


269 


278 


286 


295 


303 


312 321 




5oS 


329 


338 


346 


355 


364 


372 


38i 


389 


398 4° 6 


506 


415 


424 


432 


441 


449 


458 


467 


475 


484 492 


9 


507 


501 


509 


518 


526 


535 


544 


552 


5^1 


5 6 9 578 




508 


586 


595 


603 


612 


621 


629 


638 


646 


655 663 




°/9 
1,8 

3,6 
4,5 
5,4 
6,3 

7,2 

8,1 


5°9 


672 


680 


689 


697 


706 


714 


723 


73 1 


740 749 


3 
4 
5 
6 

7 
8 


510 


757 


766 


774 


783 


791 


800 


808 


817 


825 834 


511 


842 


851 


859 


868 


876 


885 


893 


902 


910 919 


S12 


927 


935 


944 


952 


961 


969 


978 


986 


995 #003 


513 


71 012 


020 


029 


o37 


046 


o54 


063 


071 


079 088 


5M 


096 


105 


113 


122 


130 


139 


147 


155 


164 172 


9 


5i5 


181 


189 


198 


206 


214 


223 


231 


240 


248 257 




516 


265 


273 


282 


290 


299 


307 


315 


324 


332 34i 




5i7 


349 


357 


36b 


374 


383 


39i 


399 


408 


416 425 




5i» 


433 


441 


45o 


458 


466 


475 


483 


492 


500 508 




519 


517 


525 


533 


542 


55o 


559 


567 


575 


584 592 




520 


600 


609 


617 


625 


634 


642 


650 


659 


667 675 


521 


684 


692 


700 


709 


717 


725 


734 


742 


75o 759 


8 


522 


767 


775 


784 


792 


800 


809 


817 


82^ 


834 842 


523 


850 


858 


867 


875 


883 


892 


900 


908 


917 925 


1 


1,6 

2,4 
3,2 


524 


933 


941 


95o 


958 


966 


975 


983 


991 


999 #008 


2 

3 

4 


525 


72 016 


024 


032 


041 


049. 


o57 


066 


074 


082 090 


526 


099 


107 


115 


123 


132 


140 


148 


156 


165 173 


5 
6 


4,° 
4,8 
5,6 
6,4 

7/2 


527 


181 


189 


198 


206 


214 


222 


230 


239 


247 255 


528 


263 


272 


280 


288 


296 


3°4 


313 


321 


329 337 


7 
8 

9 


529 


346 


354 


362 


370 


378 


387 


395 


403 


411 419 


530 


428 


436 


444 


4^2 


460 


469 


477 


485 


493 5oi 




S3i 


509 


5i8 


526 


534 


542 


55o 


558 


567 


575 583 




532 


59i 


S99 


607 


616 


624 


632 


640 


648 


656 665 




533 


673 


681 


689 


697 


705 


713 


722 


730 


738 746 




534 
535 


754 


762 


770 


779 


787 


795 


803 


811 


819 827 




835 


843 


852 


860 


868 


876 


884 


892 


900 908 


536 


91b 


925 


933 


941 


949 


957 


9^5 


973 


981 989 


7 


537 


997 


*oo6 


3,014 *022 ^030 


*Q38 


*Q4b 


*°54 


#062 ^070 


53« 


73078 


086 


094 


102 


in 


119 


127 


135 


143 151 


1 


o,7 


539 


159 


167 


175 


183 


191 


199 


207 


215 


223 231 


2 

3 
4 


i,4 
2,1 
2,8 


540 


239 


247 


2^S 


263 


272 


280 


288 


296 


304 312 


54i 


320 


328 


336 


344 


352 


360 


368 


376 


384 392 


5 
6 


3,5 


542 


400 


408 


416 


424 


432 


440 


448 


456 


464 472 


4,2 


543 


480 


488 


4 q6 


^04 


S12 


520 


S28 


53 6 


544 552 


7 
8 

9 


4,9 
5,6 
6,3 


544 


560 


568 


576 


584 


592 


600 


608 


61b 


624 632 


545 


640 


648 


656 


664 


672 


679 


687 


695 


703 711 




546 


719 


727 


73^ 


743 


7Si 


759 


767 


775 


783 791 




547 


799 


807 


815 


823 


830 


838 


846 


854 


862 870 




548 


878 


886 


894 


902 


910 


918 


926 


933 


941 949 




549 


957 


965 


973 


981 


989 


997 


*005 *OI3 *020 *028 




550 


74036 


044 


052 


060 


068 


076 


084 


092 


099 107 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


550 


74036 


044 


052 


060 


068 


076 


084 


092 


099 


107 




5Si 


ii5 


123 


131 


i39 


147 


i55 


162 


170 


178 


186 




55 2 


194 


202 


210 


218 


225 


233 


241 


249 


257 


265 




553 


273 


280 


288 


296 


3°4 


312 


320 


3 2 7 


335 


343 




554 

555 


35i 


359 


367 


374 


382 


39° 


398 


406 


414 


421 




429 


437 


445 


453 


461 


468 


476 


484 


492 


500 


556 


507 


515 


523 


53i 


539 


547 


554 


5° 2 


570 


578 




557 


586 


593 


601 


609 


617 


624 


632 


640 


648 


656 




558 


663 


671 


679 


687 


695 


702 


710 


718 


726 


733 




559 


74i 


749 


757 


764 


772 


780 


788 


796 


803 


811 




560 


819 


827 


834 


842 


850 


858 


865 


873 


881 


889 


561 


896 


904 


912 


920 


927 


935 


943 


95o 


958 


966 


8 


562 


974 


981 


989 


997 *°°5 


*° 12 


*020 


*028 


*°35 


*°43 


j /-> a 


563 


75 051 


059 


066 


074 


082 


089 


O97 


105 


113 


120 




i^6 

2,4 
3,2 
4,o 
4,8 
5,6 
6,4 
7,2 


564 


128 


136 


143 


151 


159 


166 


174 


182 


189 


197 


3 
4 
5 
6 


565 


205 


213 


220 


228 


236 


243 


251 


259 


266 


274 


566 


282 


289 


297 


305 


312 


320 


328 


335 


343 


35i 


567 


358 


366 


374 


38i 


389 


397 


404 


412 


420 


427 


7 
8 


568 


435 


442 


450 


458 


465 


473 


481 


488 


496 


504 


569 


5ii 


519 


526 


534 


542 


549 


557 


565 


572 


580 


9 


570 


587 


595 


603 


610 


618 


626 


633 


641 


648 


656 




57i 


664 


671 


679 


686 


694 


702 


709 


717 


724 


732 




572 


740 


747 


755 


762 


770 


778 


785 


793 


800 


808 




573 


8i5 


823 


831 


838 


846 


853 


861 


868 


876 


884 




574 


891 


899 


906 


914 


921 


929 


937 


944 


952 


959 




575 


967 


974 


982 


989 


997 


*oo5 *oi2 *020 3.027 3,033 


576 


76 042 


050 


o57 


063 


072 


080 


087 


095 


103 


no 




577 


118 


125 


133 


140 


148 


155 


163 


170 


178 


i85 




578 


193 


200 


208 


215 


223 


230 


238 


245 


253 


260 




579 


268 


275 


283 


290 


298 


305 


313 


320 


328 


335 




580 


343 


35° 


358 


365 


373 


380 


388 


395 


403 


410 


581 


418 


425 


433 


440 


448 


455 


462 


470 


477 


483 


7 


582 


492 


500 


507 


515 


522 


530 


537 


545 


552 


559 




583 


567 


574 


582 


589 


597 


604 


612 


619 


626 


634 


1 0,7 


584 


641 


649 


656 


664 


671 


678 


686 


693 


701 


708 


2 1,4 

3 2,1 

4 2,8 

5 3,5 

6 4,2 

7 4,9 

8 5,6 
916,3 


585 


716 


723 


730 


738 


745 


753 


760 


768 


775 


782 


586 


790 


797 


803 


812 


819 


827 


834 


842 


849 


856 


587 


864 


871 


879 


886 


893 


901 


908 


916 


923 


93o 


588 


938 


945 


953 


960 


967 


975 


982 


989 


997 


*oo4 


589 


'j'j 012 


019 


026 


o34 


041 


048 


056 


063 


070 


078 


590 


085 


093 


100 


107 


115 


122 


129 


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144 


151 


59i 


159 


166 


173 


181 


188 


195 


203 


210 


217 


223 




592 


232 


240 


247 


254 


262 


269 


276 


283 


291 


298 




593 


3o5 


313 


320 


327 


335 


342 


349 


357 


364 


37i 




594 


379 


386 


393 


401 


408 


4i5 


422 


430 


437 


444 




595 


452 


459 


466 


474 


481 


488 


495 


503 


5io 


517 


596 


525 


532 


539 


546 


554 


56i 


568 


576 


583 


590 




597 


597 


605 


612 


619 


627 


634 


641 


648 


656 


663 




598 


670 


677 


685 


692 


699 


706 


714 


721 


728 


735 




599 


743 


75° 


757 


764 


772 


779 


786 


793 


801 


808 




600 


815 


822 


830 


837 


844 


851 


859 


866 


873 


880 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


600 


77 8i5 


822 


830 


837 


844 


851 


859 


866 


873 


880 




601 


887 


895 


902 


909 


916 


924 


93i 


938 


945 


952 




602 


960 


967 


974 


981 


988 


996 *oo3 *oio *oi7 *025~ 




603 


78032 


039 


046 


053 


061 


068 


075 


082 


089 


097 




604 


104 


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118 


125 


132 


140 


147 


154 


161 


168 




605 


176 


183 


190 


197 


204 


211 


219 


226 


233 


240 


606 


247 


254 


262 


269 


276 


283 


290 


297 


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312 


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607 


319 


326 


333 


34° 


347 


355 


362 


3 6 9 


376 


383 




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608 


39° 


398 


405 


412 


419 


426 


433 


440 


447 


455 


1 


609 


462 


469 


476 


483 


490 


497 


5°4 


512 


519 


526 


2 
3 

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5 
6 


610 


533 


540 


547 


554 


56i 


569 


576 


583 


590 


597 


611 


604 


611 


618 


625 


633 


640 


647 


654 


661 


668 


4/° 

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612 


675 


682 


689 


696 


704 


711 


718 


725 


73 2 


739 


7 
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7/2 


613 


746 


753 


760 


767 


774 


781 


789 


796 


803 


810 


614 


817 


824 


831 


838 


845 


852 


859 


866 


873 


880 


9 


615 


888 


895 


902 


909 


916 


923 


930 


937 


944 


95i 




616 


958 


965 


972 


979 


986 


993 *°°° *°°7 *oi4 #021 




617 


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036 


043 


050 


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064 


071 


078 


085 


092 




618 


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106 


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120 


127 


134 


141 


148 


155 


162 




619 


169 


176 


183 


190 


197 


204 


211 


218 


225 


232 




620 


239 


246 


253 


260 


267 


274 


281 


288 


295 


302 


621 


309 


316 


323 


330 


337 


344 


35i 


358 


365 


372 


7 


622 


379 


386 


393 


400 


407 


414 


421 


428 


435 


442 




623 


449 


456 


463 


470 


477 


484 


491 


498 


5o5 


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624 


518 


525 


532 


539 


546 


553 


560 


567 


574 


58i 


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3 
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1,4 
2,1 
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625 


588 


595 


602 


609 


616 


623 


630 


637 


644 


650 


626 


657 


664 


671 


678 


685 


692 


699 


706 


713 


720 


5 
6 


627 


727 


734 


74i 


748 


754 


761 


768 


775 


782 


789 


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628 


796 


803 


810 


817 


824 


831 


837 


844 


851 


858 


7 
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4,9 
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629 


865 


872 


879 


886 


893 


900 


906 


913 


920 


927 


9 


630 


934 


941 


948 


955 


962 


969 


975 


982 


989 


996 




631 


80003 


010 


017 


024 


030 


037 


044 


051 


058 


065 




632 


072 


079 


085 


092 


099 


106 


113 


120 


127 


J 34 




633 


140 


147 


154 


161 


168 


175 


182 


188 


195 


202 




634 


209 


216 


223 


229 


236 


243 


250 


257 


264 


271 




635 


277 


284 


291 


298 


3°5 


312 


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325 


332 


339 


636 


346 


353 


359 


366 


373 


380 


387 


393 


400 


407 


6 


637 


414 


421 


428 


434 


441 


448 


455 


462 


468 


475 




638 


482 


489 


496 


502 


509 


5i6 


523 


53° 


536 


543 


1 


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639 


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557 


564 


570 


577 


584 


59i 


598 


604 


611 


2 

3 
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640 


618 


625 


632 


638 


645 


652 


659 


665 


672 


679 


641 


686 


693 


699 


706 


713 


720 


726 


733 


740 


747 


5 
6 


3/° 

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642 


754 


760 


767 


774 


781 


787 


794 


801 


808 


814 


643 


821 


828 


835 


841 


848 


855 


862 


868 


875 


882 


7 
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9 


4,2 
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644 


889 


895 


902 


909 


916 


922 


929 


936 


943 


949 


645 


956 


963 


969 


976 


983 


990 


996 *oo3 *oio #017 




646 


81 023 


030 


037 


043 


050 


057 


064 


070 


077 


084 




647 


090 


097 


104 


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117 


124 


131 


137 


144 


151 




648 


158 


164 


171 


178 


184 


191 


198 


204 


211 


218 




649 


224 


231 


238 


245 


251 


258 


265 


271 


278 


285 




650 


291 


298 


3o5 


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325 


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338 


345 


351 


N. 


L. 


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5 


6 


7 


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P.P. 



N. 


L. 


1 


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3 


4 


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6 


7 


8 


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P.P. 


650 


81 291 


298 


305 


311 


318 


323 


33i 


338 


343 


35i 




651 


358 


365 


37i 


378 


383 


39i 


398 


403 


411 


418 




652 


42? 


43i 


438 


443 


45i 


458 


463 


471 


478 


483 




653 


491 


498 


5o5 


5ii 


518 


523 


53i 


538 


544 


55i 




654 


558 


5°4 


57i 


578 


584 


59i 


598 


604 


611 


617 




655 


624 


631 


637 


644 


651 


657 


664 


671 


677 


684 


656 


690 


697 


704 


710 


717 


723 


730 


737 


743 


750 




657 


757 


763 


770 


776 


783 


790 


796 


803 


809 


816 




658 


823 


829 


836 


842 


849 


856 


862 


869 


875 


882 




659 


889 


895 


902 


908 


9*5 


921 


928 


935 


941 


948 




660 


954 


961 


968 


974 


981 


987 


994 *ooo *oc>7 *oi4 


661 


82 020 


027 


033 


040 


046 


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060 


066 


073 


079 


7 


662 


086 


092 


099 


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112 


119 


125 


132 


138 


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0,7 


663 


151 


158 


164 


171 


178 


184 


191 


197 


204 


210 


664 


217 


223 


230 


236 


243 


249 


256 


263 


269 


276 


2 

3 

4 
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6 


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2,1 
2,8 

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6,3 


665 


282 


289 


295 


302 


308 


3i3 


321 


328 


334 


34i 


666 


347 


354 


360 


367 


373 


380 


387 


393 


400 


406 


667 


413 


419 


426 


432 


439 


445 


452 


458 


463 


47i 


7 
8 


668 


478 


484 


491 


497 


504 


5io 


517 


523 


53o 


536 


669 


543 


549 


556 


562 


569 


575 


582 


588 


593 


601 


9 


670 


607 


614 


620 


627 


633 


640 


646 


653 


659 


666 




671 


672 


679 


685 


692 


698 


7o3 


711 


718 


724 


73° 




672 


737 


743 


75° 


756 


763 


769 


776 


782 


789 


795 




673 


802 


808 


814 


821 


827 


834 


840 


847 


853 


860 




674 


866 


872 


879 


885 


892 


898 


9o3 


911 


918 


924 




675 


93° 


937 


943 


95o 


956 


963 


969 


975 


982 


988 


676 


993 *°°i 


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#OI4 #020 


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^046 


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677 


83 059 


065 


072 


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085 


091 


097 


104 


no 


117 




678 


123 


129 


136 


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149 


153 


161 


168 


174 


181 




679 


187 


193 


200 


206 


213 


219 


225 


232 


238 


243 




680 


251 


257 


264 


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276 


283 


289 


296 


302 


308 


681 


315 


321 


327 


334 


340 


347 


353 


359 


366 


372 


6 


682 


378 


385 


39i 


398 


404 


410 


417 


423 


429 


436 




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683 


442 


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455 


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467 


474 


480 


487 


493 


499 


1 


684 


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512 


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537 


544 


550 


556 


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685 


569 


575 


582 


588 


594 


601 


607 


613 


620 


626 


686 


632 


639 


645 


651 


658 


664 


670 


677 


683 


689 


687 


696 


702 


708 


7i3 


721 


727 


734 


740 


746 


753 


688 


759 


765 


771 


778 


784 


790 


797 


803 


809 


816 


689 


822 


828 


833 


841 


847 


853 


860 


866 


872 


879 


690 


885 


891 


897 


904 


910 


916 


923 


929 


935 


942 




691 


948 


954 


960 


967 


973 


979 


985 


992 


998 


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693 


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080 


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092 


098 


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117 


123 


130 




694 


136 


142 


148 


153 


161 


167 


173 


180 


186 


192 




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198 


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211 


217 


223 


230 


236 


242 


248 


253 


696 


261 


267 


273 


280 


286 


292 


298 


305 


3ii 


317 




697 


323 


33° 


336 


342 


348 


354 


361 


3 6 7 


373 


379 




698 


386 


392 


398 


404 


410 


417 


423 


429 


435 


442 




699 


448 


454 


460 


466 


473 


479 


483 


491 


497 


5°4 




700 


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528 


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54i 


547 


553 


559 


566 


N. 


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1 


2 


3 


4 


5 


6 


7 


8 


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P.P. 



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L. 


1 


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6 


7 


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P.P. 


700 


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528 


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547 


553 


559 


566 




701 


572 


578 


584 


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597 


603 


609 


615 


621 


628 




702 


634 


640 


646 


652 


658 


665 


671 


677 


683 


689 




703 


696 


702 


708 


714 


720 


726 


733 


739 


745 


75i 




704 


757 


763 


770 


776 


782 


788 


794 


800 


807 


813 




705 


819 


825 


831 


837 


844 


850 


856 


862 


868 


874 


706 


880 


887 


893 


899 


905 


911 


917 


924 


930 


936 


7 


707 


942 


948 


954 


960 


967 


973 


979 


985 


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708 


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016 


022 


028 


034 


040 


046 


052 


058 


709 


065 


071 


077 


083 


089 


095 


101 


107 


114 


120 


3 
4 

5 


710 


126 


132 


138 


144 


150 


156 


163 


169 


175 


181 


711 


187 


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199 


205 


211 


217 


224 


230 


236 


242 


6 


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712 


248 


254 


260 


266 


272 


278 


285 


291 


297 


303 


7 


713 


309 


315 


321 


327 


333 


339 


345 


352 


358 


364 


8 


5,6 


714 


37o 


376 


382 


388 


394 


400 


406 


412 


418 


425 


9 


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43i 


437 


443 


449 


455 


461 


467 


473 


479 


485 




716 


491 


497 


503 


509 


5i6 


522 


528 


534 


540 


546 




717 


552 


558 


504 


570 


576 


582 


588 


594 


600 


606 




718 


612 


6i3 


625 


631 


637 


643 


649 


655 


661 


667 




719 
720 


673 


679 


685 


691 


697 


703 


709 


715 


721 


727 




733 


739 


745 


751 


757 


763 


769 


775 


781 


788 


721 


794 


800 


806 


812 


818 


824 


830 


836 


842 


848 


6 


722 


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860 


866 


872 


878 


884 


890 


896 


902 


908 


1 o,6 

2 1,2 

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723 


914 


920 


926 


932 


938 


944 


950 


956 


962 


968 


724 


974 


980 


986 


992 


998 


#004 #010 


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725 


86 034 


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058 


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5 
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726 


094 


100 


106 


112 


118 


124 


130 


136 


I 4 I 


147 


727 


153 


159 


165 


171 


177 


183 


189 


195 


20I 


207 


7 
8 


728 


213 


219 


225 


231 


237 


243 


249 


255 


261 


267 


729 


273 


279 


285 


291 


297 


303 


308 


314 


320 


326 


9 


730 


332 


338 


344 


35o 


356 


362 


368 


374 


380 


386 




73i 


392 


398 


404 


410 


415 


421 


427 


433 


439 


445 




732 


45i 


457 


463 


469 


475 


481 


487 


493 


499 


504 




733 


5io 


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528 


534 


540 


546 


552 


558 


564 




734 


570 


576 


58i 


587 


593 


599 


605 


611 


617 


623 




735 


629 


635 


641 


646 


652 


658 


664 


670 


676 


682 


736 


688 


694 


700 


705 


711 


717 


723 


729 


735 


741 


5 


737 


747 


753 


759 


764 


770 


776 


782 


788 


794 


800 




738 


806 


812 


817 


823 


829 


835 


841 


847 


853 


859 


1 


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739 


864 


870 


876 


882 


888 


894 


900 


906 


911 


917 


2 

3 
4 
5 
6 


1 1,0 

I 1 ' 5 

2,0 

i 2 ,5 
3,o 
3,5 
4,o 
4,5 


740 


923 


929 


935 


941 


947 


953 


958 


964 


970 


976 


741 


982 


988 


994 


999 *oo5 


*OII 


*oi7 *023 *029 *<>35 


742 


87 040 


046 


052 


058 


064 


070 


075 


081 


087 


093 


7 
8 


743 


099 


105 


in 


116 


122 


128 


134 


140 


146 


151 


744 


157 


163 


169 


175 


181 


186 


192 


198 


204 


210 


9 


745 


216 


221 


227 


233 


239 


245 


251 


256 


262 


268 




746 


274 


280 


286 


291 


297 


303 


309 


315 


320 


326 




747 


332 


338 


344 


349 


355 


361 


367 


373 


379 


384 




748 


390 


396 


402 


408 


413 


419 


425 


43i 


437 


442 




749 


448 


454 


460 


466 


471 


477 


483 


489 


495 


500 




750 


506 


512 


5i8 


523 


529 


535 


54i 


547 


552 


558 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


750 


87 506 


512 


518 


523 


529 


535 


54i 


547 


552 


558 




7Si 


5 6 4 


570 


576 


581 


587 


593 


599 


604 


610 


616 




752 


622 


628 


633 


639 


645 


651 


656 


662 


668 


674 




753 


679 


685 


691 


697 


703 


708 


714 


720 


726 


731 




754 


737 


743 


749 


754 


760 


766 


772 


777 


783 


789 




755 


795 


800 


806 


812 


818 


823 


829 


835 


841 


846 


756 


852 


858 


864 


869 


875 


881 


887 


892 


898 


904 




757 


910 


9i5 


921 


927 


933 


938 


944 


950 


955 


961 




758 


967 


973 


978 


984 


990 


996 


*ooi 


*cx>7 


*°I3 


*oi8 




759 


88 024 


030 


036 


041 


047 


o53 


058 


064 


070 


076 




760 


081 


087 


°93 


098 


104 


no 


116 


121 


127 


133 


761 


138 


144 


150 


156 


161 


167 


T-73 


178 


184 


190 


6 


762 


195 


201 


207 


213 


218 


224 


230 


235 


241 


247 


T ~ ^ 


763 


252 


258 


264 


270 


275 


281 


287 


292 


298 


304 


1 


1,2 

i,8 

2,4 

3,° 
3,6 
4,2 
4,8 
5,4 


764 


309 


3i5 


321 


326 


332 


338 


343 


349 


355 


360 


3 
4 
5 
6 


765 


366 


372 


377 


383 


389 


395 


400 


406 


412 


417 


766 


423 


429 


434 


440 


446 


45i 


457 


463 


468 


474 


767 


480 


485 


491 


497 


5° 2 


508 


513 


519 


525 


530 


7 
8 


768 


536 


542 


547 


553 


559 


564 


570 


576 


S81 


587 


769 


593 


598 


604 


610 


6i5 


621 


627 


632 


638 


643 


9 


770 


649 


655 


660 


666 


672 


677 


683 


689 


694 


700 




771 


705 


711 


717 


722 


728 


734 


739 


745 


75o 


756 




772 


762 


767 


773 


779 


784 


790 


795 


801 


807 


812 




773 


818 


824 


829 


835 


840 


846 


852 


857 


863 


868 




774 


874 


880 


885 


891 


897 


902 


908 


913 


919 


925 




775 


93o 


936 


941 


947 


953 


958 


964 


969 


975 


981 


776 


986 


992 


997 *o°3 *°°9 


*oi4 *020 *025 ^031 3,037 




777 


89 042 


048 


o53 


059 


064 


070 


076 


081 


087 


092 




778 


098 


104 


109 


US 


120 


126 


131 


137 


143 


148 




779 


154 


159 


165 


170 


176 


182 


187 


193 


198 


204 




780 


209 


215 


221 


226 


232 


237 


243 


248 


254 


260 


781 


265 


271 


276 


282 


287 


293 


298 


304 


310 


3i5 


5 


782 


321 


326 


332 


337 


343 


348 


354 


360 


365 


37i 




783 


376 


382 


387 


393 


398 


404 


409 


4i5 


421 


426 


1 


u ,5 


784 


432 


437 


443 


448 


454 


459 


465 


470 


476 


481 


2 

3 

4 

5 
6 


i,o 

i,5 

2 ,° 


785 


487 


492 


498 


504 


5°9 


515 


520 


526 


53i 


537 


786 


542 


548 


553 


559 


564 


570 


575 


58i 


586 


592 


2,5 
3,0 

3,5 
4,o 
4,5 


787 


597 


603 


609 


614 


620 


625 


631 


636 


642 


647 


7 
8 


788 


653 


658 


664 


669 


675 


680 


686 


691 


697 


702 


789 


708 


713 


719 


724 


730 


735 


741 


746 


752 


757 


9 


790 


763 


768 


774 


779 


785 


790 


796 


801 


807 


812 




791 


818 


823 


829 


834 


840 


845 


851 


856 


862 


867 




792 


873 


878 


883 


889 


894 


900 


905 


911 


916 


922 




793 


927 


933 


938 


944 


949 


955 


960 


966 


971 


977 




794 


982 


988 


993 


998 


*oo4 


3,009 *oi5 *020 


*026 


*o3i 




795 


90037 


042 


048 


053 


o59 


064 


069 


075 


080 


086 


796 


091 


097 


102 


108 


113 


119 


124 


129 


135 


140 




797 


146 


151 


157 


162 


168 


173 


179 


184 


189 


195 




798 


200 


206 


211 


217 


222 


227 


233 


238 


244 


249 




799 


255 


260 


266 


271 


276 


282 


287 


293 


298 


304 




800 


309 


314 


320 


325 


331 


336 


342 


347 


352 


358 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


800 


90309 


3i4 


320 


325 


33i 


336 


342 


347 


352 


358 




801 


363 


3 6 9 


374 


380 


385 


390 


396 


401 


407 


412 




802 


417 


423 


428 


434 


439 


445 


45o 


455 


461 


466 




803 


472 


477 


482 


488 


493 


499 


504 


509 


5i5 


520 




804 


526 


531 


536 


542 


547 


553 


558 


563 


569 


574 




805 


580 


585 


590 


596 


601 


607 


612 


617 


623 


628 


806 


634 


639 


644 


650 


655 


660 


666 


671 


677 


682 




807 


687 


693 


698 


703 


709 


714 


720 


725 


73° 


736 




808 


741 


747 


752 


757 


763 


768 


773 


779 


784 


789 




809 


795 


800 


806 


811 


816 


822 


827 


832 


838 


843 




810 


849 


854 


859 


865 


870 


875 


881 


886 


891 


897 


811 


902 


907 


9 J 3 


918 


924 


929 


934 


940 


945 


95o 


6 


812 


956 


961 


966 


972 


977 


982 


988 


993 


998 


*°04 


1 


0,6 


813 


91 009 


014 


020 


025 


030 


036 


041 


046 


052 


o57 


814 


062 


068 


073 


078 


084 


089 


094 


100 


i°5 


no 


2 
3 
4 

5 
6 


2,4 

3,<> 

3,6 
4,2 
4,8 
5/4 


815 


116 


121 


126 


132 


137 


142 


148 


153 


158 


164 


816 


169 


174 


180 


185 


190 


196 


201 


206 


212 


217 


817 


222 


228 


233 


238 


243 


249 


254 


259 


265 


270 


7 
8 


818 


275 


281 


286 


291 


297 


302 


307 


312 


3i8 


323 


819 


328 


334 


339 


344' 


350 


355 


360 


365 


37i 


376 


9 


820 


38i 


387 


392 


397 


403 


408 


413 


418 


424 


429 




821 


434 


440 


445 


450 


455 


461 


466 


471 


477 


482 




822 


487 


492 


498 


503 


508 


514 


519 


524 


529 


535 




823 


540 


545 


55i 


556 


561 


566 


572 


577 


582 


587 




824 


593 


598 


603 


609 


614 


619 


624 


630 


635 


640 




825 


645 


651 


656 


661 


666 


672 


677 


682 


687 


693 


826 


698 


703 


709 


714 


719 


724 


73° 


735 


740 


745 




827 


75i 


75 6 


761 


766 


772 


777 


782 


787 


793 


798 




828 


803 


808 


814 


819 


824 


829 


834 


840 


845 


85o 




829 


855 


861 


866 


871 


876 


882 


887 


892 


897 


903 




830 


908 


913 


918 


924 


929 


934 


939 


944 


95o 


955 


831 


960 


965 


971 


976 


981 


986 


991 


997 


#002 3,007 


5 


832 


92 012 


018 


023 


028 


o33 


038 


044 


049 


054 


o59 






833 


065 


070 


075 


080 


085 


091 


096 


101 


I06 


in 


1 


°,5 


834 


117 


122 


127 


132 


137 


143 


148 


153 


158 


163 


2 
3 
4 
5 
6 


i,o 

i/5 

2,0 
2,5 
3,o 
3,5 
4,o 
4,5 


835 


169 


174 


179 


184 


189 


195 


200 


205 


2IO 


215 


836 


221 


226 


231 


236 


241 


247 


252 


257 


262 


267 


837 


273 


278 


283 


288 


293 


298 


304 


309 


314 


3*9 


7 
8 


838 


324 


33° 


335 


340 


345 


350 


355 


361 


366 


37i 


839 


376 


38i 


387 


392 


397 


402 


407 


412 


418 


423 


9 


840 


428 


433 


438 


443 


449 


454 


459 


464 


469 


474 




841 


480 


485 


490 


495 


500 


505 


5ii 


5i6 


521 


526 




842 


53i 


536 


542 


547 


552 


557 


562 


567 


572 


-578 




843 


583 


588 


593 


598 


603 


609 


614 


619 


624 


629 




844 


634 


639 


645 


650 


655 


660 


665 


670 


675 


681 




845 


686 


691 


696 


701 


706 


711 


716 


722 


727 


73 2 


846 


737 


742 


747 


752 


758 


763 


768 


773 


778 


783 




847 


788 


793 


799 


804 


809 


814 


819 


824 


829 


834 




848 


840 


845 


850 


855 


860 


865 


870 


875 


88l 


886 




849 


891 


896 


901 


906 


911 


916 


921 


927 


932 


937 




850 


942 


947 


952 


957 


962 


967 


973 


978 


983 


988 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


850 


92942 


947 


952 


957 


962 


967 


973 


978 


983 


988 




851 


993 


998 


*oo 3 


*oo8 


*oi3 


*oi8 


*024 3,029 *034 3,039 




852 


93°44 


049 


o54 


059 


064 


069 


075 


080 


085 


090 




853 


095 


100 


105 


no 


"5 


120 


125 


131 


136 


141 




854 


146 


151 


156 


161 


166 


171 


176 


181 


186 


192 




855 


197 


202 


207 


212 


217 


222 


227 


232 


237 


242 


856 


247 


252 


258 


263 


268 


273 


278 


283 


288 


293 


6 


857 


298 


3°3 


308 


313 


318 


323 


328 


334 


339 


344 


j ^ a 


858 


349 


354 


359 


364 


369 


374 


379 


384 


389 


394 


3 

4 
5 
6 


1,2 

1,8 

2,4 

3,o 
3-6 

4,2 

4,8 
5,4 


859 


399 


404 


409 


4I4 


420 


425 


43o 


435 


440 


445 


860 


45° 


455 


460 


465 


47O 


475 


480 


485 


490 


495 


861 


500 


5°5 


5io 


515 


520 


526 


531 


536 


54i 


546 


862 


55i 


556 


56i 


566 


571 


576 


58i 


586 


59i 


596 


7 

8 


863 


601 


606 


611 


6l6 


621 


626 


631 


636 


641 


646 


864 


651 


656 


661 


666 


67I 


676 


682 


687 


692 


697 


9i 


865 


702 


707 


712 


717 


722 


727 


732 


737 


742 


747 




866 


752 


757 


762 


767 


772 


m 


782 


787 


792 


797 




867 


802 


807 


812 


817 


822 


Zorj 


832 


837 


842 


847 




868 


852 


857 


862 


867 


872 


877 


882 


887 


892 


897 




869 
870 


902 


907 


912 


917 


922 


927 


932 


937 


942 


947 


• 


952 


957 


962 


967 


972 


977 


982 


987 


992 


997 


871 


94 002 


007 


012 


017 


022 


027 


032 


037 


042 


047 


a 


872 


052 


o57 


062 


067 


072 


077 


082 


086 


091 


096 






873 


101 


106 


in 


116 


121 


126 


131 


136 


141 


146 


1 


°,5 


874 


151 


156 


161 


166 


171 


176 


181 


186 


191 


196 


2 
3 
4 


I ,° 
i,5 
2,0 
2,5 
3,o 
3,5 
4,° 
4,5 


875 


201 


206 


211 


216 


221 


226 


231 


236 


240 


245 


876 


250 


255 


260 


265 


27O 


275 


280 


285 


290 


295 


5 

6 


877 


300 


3°5 


310 


315 


320 


325 


33° 


335 


34° 


345 


7 
8 


878 


349 


354 


359 


364 


3 6 9 


374 


379 


384 


389 


394 


879 


399 


404 


409 


414 


419 


424 


429 


433 


438 


443 


9 


880 


448 


453 


458 


463 


468 


473 


478 


483 


488 


493 




881 


498 


503 


507 


512 


517 


522 


527 


532 


537 


542 




882 


547 


552 


557 


562 


567 


57i 


576 


58i 


586 


591 




883 


596 


601 


606 


611 


616 


621 


626 


630 


635 


640 




884 


645 


650 


655 


660 


665 


670 


675 


680 


685 


689 




885 


694 


699 


704 


709 


714 


719 


724 


729 


734 


738 


886 


743 


748 


753 


758 


763 


768 


773 


778 


783 


787 


4 


887 


792 


797 


802 


807 


812 


817 


822 


827 


832 


836 


888 


841 


846 


851 


856 


861 


866 


871 


876 


880 


885 


1 


o,4 


889 


890 


895 


900 


905 


910 


915 


919 


924 


929 


934 


2 
3 

4 
5 
6 


°,8 

1,2 

1,6 
2,0 


890 


939 


944 


949 


954 


959 


963 


968 


973 


978 


983 


891 


988 


993 


998 


*002 *007 


#012 3,017 


*022 


3,027 ^032 


892 


95 36 


041 


046 


051 


056 


061 


066 


07I 


075 


080 


2,4 
2,8 


893 


085 


090 


095 


100 


105 


109 


114 


119 


124 


129 


7 
8 

9 


894 


134 


139 


143 


148 


153 


158 


163 


168 


173 


177 


3,2 

3,6 


895 


182 


187 


192 


197 


202 


207 


211 


2l6 


221 


226 




896 


231 


236 


240 


245 


250 


255 


260 


265 


27O 


274 




897 


279 


284 


289 


294 


299 


303 


308 


313 


318 


323 




898 


328 


332 


337 


342 


347 


352 


357 


361 


366 


37i 




899 


376 


38i 


386 


390 


395 


400 


4o5 


4IO 


415 


419 




900 


424 


429 


434 


439 


444 


448 


453 


458 


463 


468 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


900 


95424 


429 


434 


439 


444 


448 


453 


458 


463 


468 




901 


472 


477 


482 


487 


492 


497 


501 


506 


511 


5io 




902 


521 


525 


530 


535 


54o 


545 


5 So 


554 


559 


504 




903 


569 


574 


578 


583 


588 


593 


S98 


602 


607 


612 




904 


617 


622 


626 


631 


636 


641 


646 


650 


655 


660 




9°S 


665 


670 


674 


679 


684 


689 


694 


698 


703 


708 


906 


713 


718 


722 


727 


732 


737 


742 


746 


75i 


756 




907 


761 


766 


770 


775 


780 


78s 


789 


794 


799 


804 




908 


809 


813 


818 


823 


828 


832 


837 


842 


847 


8^2 




909 


856 


861 


866 


8 7 i 


875 


880 


885 


890 


895 


899 




910 


904 


909 


914 


918 


923 


928 


933 


938 


942 


947 


911 


952 


957 


961 


966 


971 


976 


980 


985 


990 


995 


5 


912 


999 *o°4 


*oo9 #014 *oi9 


*023 


#028 


*Q33 


^.038 3,042 


1 


°/5 


913 


96047 


052 


057 


061 


066 


071 


076 


080 


085 


090 


2 


I /° 


914 


095 


099 


104 


109 


114 


118 


123 


128 


133 


137 


3 

4 
5 


i,5 

2 /0 

2,5 


915 


142 


147 


152 


156 


161 


166 


171 


175 


180 


185 


916 


190 


194 


199 


204 


209 


213 


218 


223 


227 


232 


6 


3,0 


917 


237 


242 


246 


251 


256 


261 


26 s 


270 


275 


280 


7 


3,5 


918 


284 


289 


294 


298 


303 


308 


313 


317 


322 


327 


8 


4,o 


919 


332 


330 


34i 


340 


'350 


355 


360 


305 


309 


374 


9 


4,5 s 


920 


379 


384 


388 


393 


398 


402 


407 


412 


417 


421 




921 


426 


431 


435 


440 


445 


45o 


454 


459 


464 


468 




922 


473 


478 


483 


4«7 


492 


497 


501 


506 


5il 


515 




923 


520 


525 


530 


534 


539 


544 


548 


553 


558 


562 




924 


567 


572 


577 


58i 


586 


59i 


595 


600 


605 


609 




925 


614 


619 


624 


628 


6^3 


638 


642 


647 


6^2 


6^6 


926 


661 


666 


670 


675 


680 


685 


689 


694 


699 


703 




927 


708 


713 


717 


722 


727 


73i 


730 


741 


745 


75o 




928 


755 


7S9 


764 


769 


774 


778 


783 


788 


792 


797 




929 


802 


806 


811 


816 


820 


825 


830 


834 


839 


844 




930 


848 


853 


858 


862 


867 


872 


876 


881 


886 


890 


93i 


895 


900 


904 


909 


914 


918 


923 


928 


932 


937 


4 


932 


942 


946 


95i 


950 


960 


9o5 


970 


974 


979 


984 




0,4 
8 


933 


988 


993 


997 


*002 


*oo7 


*on 


*oi6 


#021 


*025 


*030 




934 


97035 


039 


044 


049 


053 


058 


063 


067 


072 


077 


3 
4 
5 
6 


1,2 

1,6 


935 


081 


086 


090 


oqi, 


100 


104 


109 


114 


118 


123 


936 


128 


132 


137 


142 


146 


151 


155 


160 


105 


169 


2,4 

2,8 


937 


174 


179 


183 


188 


192 


197 


202 


206 


211 


216 


7 
8 


938 


220 


225 


230 


234 


239 


243 


248 


253 


2.S7 


262 


3,2 

3,6 


939 


267 


271 


276 


280 


285 


290 


294 


299 


304 


308 


9 


940 


313 


317 


322 


327 


33i 


336 


340 


345 


35o 


354 




941 


359 


364 


368 


373 


377 


382 


3«7 


39i 


390 


400 




942 


405 


410 


414 


419 


424 


428 


433 


437 


442 


447 




943 


45i 


45o 


460 


4t>5 


470 


474 


479 


483 


488 


493 




944 

945 


497 


5°2 


506 


5ii 


5i6 


520 


525 


529 


534 


539 




543 


S48 


552 


557 


562 


566 


57i 


57.5 


580 


585 


946 


589 


S94 


598 


603 


607 


612 


617 


621 


626 


630 




947 


635 


640 


644 


649 


6S3 


658 


663 


667 


672 


676 




948 


681 


68$ 


690 


695 


699 


704 


708 


7i3 


717 


722 




949 


727 


731 


730 


74o 


745 


749 


754 


759 


763 


768 




950 


772 


777 


782 


786 


791 


795 


800 


804 


809 


813 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 


950 


97772 


m 


782 


786 


791 


795 


800 


804 


809 


813 




95i 


818 


823 


827 


832 


836 


841 


845 


850 


855 


859 




95 2 


864 


868 


873 


877 


882 


886 


891 


896 


900 


905 




953 


909 


914 


918 


923 


928 


932 


937 


941 


946 


95o 




954 


955 


959 


964 


968 


973 


978 


982 


987 


991 


996 




955 


98 000 


005 


009 


014 


019 


023 


028 


032 


037 


041 


956 


046 


050 


o55 


059 


064 


068 


073 


078 


082 


087 




957 


091 


096 


100 


105 


109 


114 


118 


123 


127 


132 




958 


137 


141 


146 


150 


155 


159 


164 


168 


173 


177 




959 


182 


186 


191 


195 


200 


204 


209 


214 


218 


223 




960 


227 


232 


236 


241 


245 


250 


254 


259 


263 


268 


961 


272 


277 


281 


286 


290 


295 


299 


304 


308 


313 


5 


962 


3i8 


322 


327 


33i 


336 


340 


345 


349 


354 


358 


1 


o,5 


963 


363 


367 


372 


376 


38i 


385 


390 


394 


399 


403 


2 


964 


408 


412 


417 


421 


426 


43o 


vss 


439 


444 


448 


3 
4 

5 


i/5 
2 /° 

2,5 


965 


453 


457 


462 


466 


47i 


475 


480 


484 


489 


493 


966 


498 


502 


507 


5* 1 


5i6 


520 


525 


529 


534 


538 


6 


3,° 


967 


543 


547 


552 


556 


56i 


565 


570 


574 


579 


583 


7 


3,5 


968 


588 


592 


597 


601 


605 


610 


614 


619 


623 


628 


8 


4,o 


969 


632 


637 


641 


646 


650 


655 


659 


664 


668 


673 


9 


4,5 


970 


677 


682 


686 


691 


695 


700 


704 


709 


713 


717 




971 


722 


726 


73i 


735 


740 


744 


749 


753 


758 


762 




972 


767 


771 


776 


780 


784 


789 


793 


798 


802 


807 




973 


811 


816 


820 


825 


829 


834 


838 


843 


847 


851 




974 


856 


860 


865 


869 


874 


878 


883 


887 


892 


896 




975 


900 


9°5 


909 


914 


918 


923 


927 


932 


936 


941 


976 


945 


949 


954 


958 


963 


967 


972 


976 


981 


985 




977 


989 


994 


998 


*oo3 


*oc>7 


*OI2 


-#°i6 


*02I 


*°25 


*029 




978 


99 034 


038 


043 


047 


052 


O56 


061 


065 


069 


074 




979 


078 


083 


087 


092 


096 


IOO 


105 


IO9 


114 


118 




980 


123 


127 


131 


136 


140 


H5 


149 


154 


158 


162 


981 


167 


171 


176 


180 


185 


189 


193 


I98 


202 


207 


4 


982 


211 


216 


220 


224 


229 


233 


238 


242 


247 


251 






983 


255 


260 


264 


269 


273 


277 


282 


286 


291 


295 


1 


°,4 
0,8 

1,2 

1,6 

2 


984 


300 


3°4 


308 


313 


317 


322 


326 


33° 


335 


339 


2 

3 

4 

5 
6 


985 


344 


348 


352 


357 


361 


366 


370 


374 


379 


383 


986 


388 


392 


396 


401 


405 


4IO 


414 


419 


423 


427 


2 A 
2 8 


987 


432 


436 


441 


445 


449 


454 


458 


463 


467 


471 


7 
8 


q88 


476 


480 


484 


489 


493 


498 


932 


506 


5ii 


5i5 


3/2 

3/6 


989 


520 


524 


528 


533 


537 


542 


546 


550 


555 


559 


9 


990 


564 


568 


572 


577 


58i 


585 


59o 


594 


599 


603 




991 


607 


612 


616 


621 


625 


629 


634 


638 


642 


647 




992 


651 


656 


660 


664 


669 


673 


677 


682 


686 


691 




993 


695 


699 


704 


708 


712 


717 


721 


726 


73° 


734 




994 


739 


743 


747 


752 


756 


760 


765 


769 


774 


778 




995 


782 


787 


791 


795 


800 


804 


808 


813 


817 


822 


996 


826 


830 


835 


839 


843 


848 


852 


856 


861 


865 




997 


870 


874 


878 


883 


887 


891 


896 


900 


904 


909 




998 


913 


917 


922 


926 


93o 


935 


939 


944 


948 


952 




999 


957 


961 


965 


970 


974 


978 


983 


987 


991 


996 




1000 


00 000 


004 


009 


013 


017 


022 


026 


030 


o35 


o39 


N. 


L. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


P.P. 



NOTES ON TABLES I AND II. 

The logarithms of numbers are in general incommensurable. 
In these tables they are given correct to five places of decimals. 
If the sixth place is 5 or more, the next larger number is used 
in the fifth place. Thus log 8102 = 3.908549+; in five-place 
tables this is written 3.90855, the dash above the 5 showing 
that the logarithm is less than given. 

So log 8133 = 3.910251- ; in five-place tables this is written 
3.91025, the dot above the 5 showing that the logarithm is more 
than given. 

In the natural functions of the angles (Table II) all numbers 
are decimals for sine and cosine (why ?), and for tangent and 
cotangent, except where the decimal point is used to indicate 
that part of the number is integral. When no decimal point 
is printed in the tables it is to be understood. When the 
natural function is a pure decimal the characteristic of the 
logarithm is negative. Accordingly, in the tables 10 is added, 
and in the result this must be allowed for. Thus 

nat. sin 44° 20' = 0.69883, log sin 44° 20' = T.84437, 

or, as printed in the tables, 9.84437, which means 9.84437 — 10. 



TABLE II. 

THE LOGARITHMIC AND NATURAL SINES, COSINES, 

TANGENTS, AND COTANGENTS OF ANGLES 

FROM 0° TO 90°. 



f 


Nat. Sin Log. 


d. 


Nat.CoSLog. 


Nat Tan Log. 


c.d. 


Log. Cot Nat. 







00000 — 




IOOOO 0.00000 


00000 — 




— 00 


60 


I 


029 6.46373 




000 0.00000 


029 6.46373 




353627 3437-7 


59 


2 


058 6.76476 


30103 
17609 
12494 
9691 
7918 
6694 
5800 


OOO 0.00000 


058 6.76470 


30103 
17609 
12494 
9691 
7918 
6694 
5800 
5ii5 
4576 
4139 
3779 
3476 
3219 
2996 
2803 
2633 
2482 
2348 


3.23524 1718.9 


58 


3 


087 6.94085 


OOO 0.00000 


087 6.94085 


3.05915 1 145.9 


57 


4 
5 


116 7.06579 


OOO 0.00000 


116 7.06579 


2.93421 859.44 


56 


00145 7.16270 


IOOOO 0.00000 


00145 7.16270 


2.83730 687.55 


55 


6 


175 7.24188 


OOO 0.00000 


175 7.24188 


2.75812 572.96 


54 


7 


204 7.30882 


000 0.00000 


204 7.30882 


2.69118 491. 11 


53 


8 


233 7-36682 


OOO 0.00000 


233 7-36682 


2.63318 429.72 


52 


9 
10 


262 7.41797 


S XI S 

4576 
4139 

3779 
3476 
3218 
2997 


OOO 0.00000 


262 7.41797 


2.58203 381.97 


5i 


00291 7.46373 


IOOOO 0.00000 


00291 7.46373 


2-53627 343-77 


50 


II 


320 7.50512 


99999 0.00000 


320 7-5°5 12 


2.49488 312.52 


49 


12 


349 7-5429I 


999 0.00000 


349 7-5429I 


2.45709 286.48 


48 


13 


378 7.57767 


999 0.00000 


378 7.57767 


2.42233 264.44 


47 


14 


407 7.60985 


999 0.00000 


407 7.60986 


2.39014 245.55 


46 
45 


15 


00436 7.63982 


99999 0.00000 


00436 7.63982 


2.36018 229.18 


ib 


465 7.66784 


2633 
2483 
2348 
2227 


999 0.00000 


465 7.66785 


2.33215 214.86 


44 


17 


495 7-694I7 


999 9-99999 


495 7-694I8 


2.30582 202.22 


43 


l8 


524 7.71900 


999 9-99999 


524 7.71900 


2.28100 190.98 


42 


19 


553 7-74248 


998 9.99999 


553 7-74248 


2.25752 180.93 


4i 
40 


20 


00582 7.76475 


99998 9.99999 


00582 7.76476 


21 19 


2.23524 171.89 


21 


611 7.78594 


9 


998 9.99999 


611 7.78595 


2.21405 163.70 


39 


22 


640 7.80615 




998 9-99999 


640 7.80615 


193 1 
1848 

1773 
1704 
1639 
1579 
1524 
1473 
1424 
1379 
1336 
1297 

1259 


2.19385 156.26 


38 


23 


669 7.82545 


1930 
1848 
1773 


998 9-99999 


669 7.82546 


2.17454 149.47 


37 


24 


698 7.8439? 


998 9-99999 


698 7.84394 


2.15606 143.24 


30 
35 


25 


00727 7.86166 


99997 9-99999 


00727 7.86167 


2.13833 I37.5I 


26 


756 7.87870 


1704 
1639 
1579 


997 9-99999 


756 7.87871 


2.12129 132.22 


34 


27 


785 7.89509 


997 9-99999 


785 7.89510 


2.10490 127.32 


33 


28 


814 7.91088 


997 9-99999 


815 7.91089 


2.0891 1 122.77 


32 


29 


844 7.92612 


I 5 2 4 
1472 


996 9.99998 


844 7.92613 


2.07387 118.54 


3i 
30 


30 


00873 7.94084 


99996 9.99998 


00873 7-94086 


2.05914 114.59 


31 


902 7.95508 


1424 


996 9.99998 


902 7-955 10 


2.04490 110.89 


29 


32 


931 7.96887 


I 379 
1336 


996 9.99998 


931 7.96889 


2.031 1 1 107.43 


28 


33 


960 7.98223 


995 9-99998 


960 7.98225 


2.01775 104.17 


27 


34 
35 


989 7.99520 


1297 
1259 


995 9-99998 


989 7-99522 


2.00478 IOI.II 


26 


01018 8.00779 


99995 9-99998 


01018 8.00781 


1.99219 98.218 


25 


36 


047 8.02002 


1223 


995 9-99998 


047 8.02004 


1223 


1.97996 95.489 


24 


37 


076 8.03192 


1158 


994 9-99997 


076 8.03194 


"9° ^96806 92.908 
1 128 i ^95647 90.463 


23 


3* 


105 8.04350 


994 9-99997 


105 8.04353 


22 


39 
40 


134 8.05478 


IIOO 


994 9-99997 


135 8.05481 


1.94519 88.144 

IIOO D „ — ~ 

1072 1 I '934i9 85.940 


21 
20 


01 164 8.06578 


99993 9-99997 


01 164 8.06581 


41 


193 8.07650 


1046 


993 9-99997 


193 8.07653 


1047 


1.92347 83.844 


19 


42 


222 8.08696 


993 9-99997 


222 8.08700 


1.91300 81.847 


18 


43 


251 8.09718 




992 9-99997 


251 8.09722 


998 
976 

955 
934 
915 
895 
878 
860 


1.90278 79.943 


i-7 


44 


280 8.10717 


999 

976 

954 


992 9-99996 


280 8.10720 


1.89280 78.126 


16 
IF 


45 


01309 8.11693 


99991 9.99996 


01309 8.11696 


1.88304 76.390 


46 


338 8.12647 


991 9.99996 


338 8.12651 


1-87349 74-729 


14 


47 


367 8.13581 


934 


991 9.99996 


367 8.13585 


1.86415 73- J 39 


13 


48 


396 8.14495 


914 
896 
877 
860 


990 9.99996 


396 8.14500 


1.85500 71.615 


12 


49 


425 8.15391 


990 9.99996 


425 8.15395 


1.84605 70.153 


11 


50 


01454 8.16268 


99989 9.99995 


01455 8.16273 


1.83727 68.750 


10 


51 


483 8.17128 


843 
827 
812 

797 
782 
769 


989 9.99995 


484 8.17133 


843 

828 


1.82867 67.402 


9 


52 


513 8.17971 


989 9.99995 


513 8.17976 


1.82024. 66.105 


8 


^3 


542 8.18798 


988 9.99995 


542 8.18804 


812 


1.81196 64.858 


7 


54 


571 8.19610 


988 9.99995 


571 8.19616 


797 
782 
769 
756 


1.80384 63.657 


6 

~5 


55 


01600 8.20407 


99987 9.99994 


01600 8.20413 


1.79587 62.499 





629 8.21 189 


987 9.99994 


629 8.21195 


1.78805 61.383 


4 


S7 


658 8.21958 


986 9-99994 


658 8.21964 


1.78036 60.306 


3 


S8 


687 8.22713 


755 


986 9.99994 


687 8.22720 


1.77280 59.266 


2 


59 


716 8.23456 


743 


985 9.99994 


716 8.23462 


742 


I-76538 58.261 


1 


60 745 8.24186 


73° 


985 9.99993 


746 8.24192 


73° 


1.75808 57.290 







Nat. COS Log 


d. 


Nat. Sin Log. 


Nat.CotLog. 


c.d. 


Log.TanNat. 


T 



89 c 



f 
~0 


Xat. Sin Log. d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 




oi745 


8.24186 ; 
8.24903 71 l 
8.25609 7°o 


99985 


9-99993 


01746 


8.24192 8 
8.24910 7I l 


1.75808 


57.290 


60 


I 


774 


984 


9-99993 


775 


1.75090 


56.351 


59 


2 


803 


984 


999993 


804 


8.25616 l ob 6 
8.26312 °9° 


1.74384 


55-442 


58 


3 


832 


8.26304 


684 

673 

663 


983 


9-99993 


833 


1.73688 


54-561 


57 


4 


862 


8.26988 


983 
99982 


9.99992 


862 


8.26996 


673 


1.73004 


53.709 


56 


5 


01891 


8.27661 


9.99992 


01891 


8.27669 


1-72331 


52.882 


55 


6 


920 


8.28324 


982 


9.99992 


920 


8.28332 1 "£ 
8.28986 , g4 
8.29629 | lf A 


1.71668 


.081 


54 


7 


949 


8.28977 


°d3 
644 
634 
624 
616 


981 


9.99992 


949 


1.71014 


51.303 


53 


8 


978 


8.29621 


980 


9.99992 


978 


1.70371 


50-549 


52 


9 
10 


02007 
02036 


8.30255 
8.30879 


980 


9.99991 


02007 


8.30263 


625 


1.69737 


49.816 


5i 
50 


99979 


9.99991 


02036 


8.30888 


1.69112 


49.104 


ii 


065 


8.3I495 


608 


979 


9.99991 


066 


8.31505 I ' 
8.32112 1 °°? 
8-327II ' g? 
8.33302 59^ 

8.33886 : \\ 


1.68495 


48.412 


49 


12 


094 


S.32103 


599 
590 

583 


978 


9.99990 


095 


1.67888 


47.740 


48 


13 


123 


8.32702 


977 


9.99990 


124 


1.67289 


.085 


47 


14 
15 


152 
02181 


8.33292 

8.33875 


977 


9.99990 


153 


1.66698 


46.449 


40 


99976 


9.99990 


02182 


1.66114 


45.829 


45 


16 


211 


8.34450 


575 
568 

--fin 


976 


9.99989 


211 


8.34461 ' HI 

8 -35029 ; £° 


165539 


.226 


44 


17 


24.0 


8.35018 


975 


9.99989 


240 


1.64971 


44-639 


43 


18 


269 


8.35578 




974 


9.99989 


269 


8.35590 ! So 


1.64410 


.066 


42 


19 
20 


298 
02327 


8.36131 
8.36678 


553 
547 


974 


9.99989 


298 


8.36143 


3DO 
546 


I-63857 


43-508 


41 
40 


99973 


9.99988 


02328 


8.36689 


1.63311 


42.964 


21 


356 


8.37217 


539 


972 


9.99988 


357 


8.37229 gr 
8.37762 fj. 
8.38289 S 


1.62771 


•433 


39 


22 


385 


8.37750 


533 

526 


972 


9.99988 


386 


1.62238 


41.916 


38 


23 


414 


8.38276 


971 


9.99987 


4i5 


1.61711 


.411 


37 


24 
25 


443 


8.38796 


520 

514 
508 


970 


9.99987 


444 


8.38809 


5H 


1.61191 


40.917 


36 


02472 


8.39310 


99969 


9.99987 


02473 


8.39323 


1.60677 


40.436 


35 


2b 


5°i 


8.39818 


969 


9.99986 


502 


8.39832 : m 


1.60168 


39-965 


34 


27 


53° 


8.40320 


502 

496 


968 


9.99986 


53i 


8.40334 ^ q6 
8.40830 496 


1.59666 


.506 


33 


28 


560 


8.40816 


967 


9.99986 


560 


1.59170 


•057 


32 


29 


5*9 


8.41307 


491 

485 
480 


966 


9.99985 


589 


8.41321 


486 


1.58679 


38.618 


3i 
30 


30 


02618 


8.41792 


99966 


9.99985 


02619 


8.41807 


i.58i93 


38.188 


31 


647 


8.42272 


965 


9.99985 


648 


8.42287 I 2;; 


I.577I3 


37.769 


29 


32 


676 


8.42746 


474 


964 


9.99984 


677 


8.42762 475 
843232 1 ff A 


I-57238 


•358 


28 


33 


705 


8.43216 


47° 
464 

459 


963 


9.99984 


706 


1.56768 


36.956 


27 


34 


734 


8.43680 


963 


9.99984 


735 


8.43696 


460 


1.56304 


•5^3 


26 


35 


02763 


8.44139 


99962 


9.99983 


02764 


8.44156 


I-55844 


36,178 


25 


3^ 


792 


8.44594 


455 


961 


9.99983 


793 


8.44611 £0 


I.55389 


35-8oi 


24 


37 


821 


8.45044 


45° 

445 


960 


9.99983 


822 


8-45061 j ^° 

8.45507 ! TT T 


1-54939 


•43i 


23 


3« 


850 


8.45489 


959 


9.99982 


851 


1-54493 


.070 


22 


39 
40 


879 


8.45930 


436 


959 


9.99982 


881 


8.45948 


437 


1.54052 


34.7!5 


21 
"20 


02908 


8.46366 


99958 


9.99982 


02910 


8.46385 


i.536i5 


34-368 


41 


938 


8.46799 


433 
427 


957 


9.99981 


939 


846817 «s 
8.47245 4 


1.53183 


.027 


19 


42 


967 


8.47226 


95° 


9.99981 


968 


I.52755 


33.694 


18 


43 


996 


8.47650 


424 


955 


9.99981 


997 


847669 424 


1-52331 


.366 


17 


44 


03025 


8.48069 


419 
416 
411 
408 

404 

400 

396 

393 
39o 
386 
382 


954 


9.99980 


03026 


8.48089 


416 


1.51911 


.045 


16 


45 


03054 


8.48485 


99953 


9.99980 


03055 


8.48505 


1. 51495 


32-73o 


15 


4 b 


083 


8.48896 


952 


9.99979 


084 


8.48917 in 

8.49325 to! 
8.49729 ft 


1.51083 


.421 


14 


47 


112 


8.49304 


952 


9.99979 


114 


1-50675 


.118 


13 


48 


141 


8.49708 


95i 


9.99979 


143 


1. 50271 


31.821 


12 


49 
50 


170 


8.50108 


95o 


9.99978 


172 


8.50130 


397 


1.49870 


.528 


11 


03199 


8.50504 


99949 


9.99978 


03201 


8.50527 


1-49473 


31.242 


10 


5i 


228 


8.50897 


948 


9.99977 


230 


8.50920 ! %* 
8.5I3IO 390 
8.51696 386 


1.49080 


30.960 


9 


52 


257 


8.51287 


947 


9.99977 


259 


1.48690 


.683 


8 


53 


286 


8.51673 


946 


9.99977 


288 


1.48304 


.412 


7 


54 
55 


316 


8.52055 


945 


9.99976 


317 


8.52079 


380 


1.47921 


•145 


6 


03345 


8.52434 


379 
376 
373 
369 
3 6 7 
363 


99944 


9.99976 


03346 


8.52459 


I4754I 


29.882 


5 


56 


374 


8.52810 


943 


9-99975 


376 


8.52835 1 J'X 
8.53208 373 

8.53578 i h° 7 
8.53945 ; i 6 l 

8.54308 3 °3 


1.47165 


.624 


4 


57 


403 


8.53183 


942 


9-99975 


405 


1.46792 


.371 


3 


5« 


432 


8.53552 


941 


9.99974 


434 


1.46422 


.122 


2 


18 


461 


8.539I9 


940 


9.99974 


463 


1.46055 


28.877 


1 


490 


8.54282 


939 


9-99974 


492 


1.45692 


.636 







Nat. COS Log. d. 


Nat. S 


in Log. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


r 



88 c 



t 


Nat. S 


in Log. d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







03490 


8.54282 


360 
357 
355 
35i 
349 
346 


99939 


9.99974 


03492 


8.54308 


361 

358 
355 
352 
349 
346 


1.45692 


28.63b 


60 


I 


5i9 


8.54642 


938 


9-99973 


521 


8.54669 


I-4533I 


•399 


59 


2 


548 


8.54999 


937 


9-99973 


550 


8.55027 


1-44973 


.166 


58 


3 


577 


8-55354 


936 


9.99972 


579 


8.55382 


1.44618 


27.937 


57 


4 
5 


bob 


8-55705 


935 


9.99972 


609 


8-55734 


1.44266 


.712 


56 


o3 6 35 


8.56054 


99934 


9.99971 


03638 


8.56083 


143917 


27.490 


55 


b 


bb4 


8.56400 


933 


9.99971 


b67 


8.56429 


I-4357I 


.271 


54 


7 


693 


8.56743 


343 
34i 


932 


9.99970 


b 9 6 


8.56773 


344 


1.43227 


•057 


53 


8 


723 


8.57084 


93i 


9.99970 


725 


8.571 14 


34 1 
338 
336 


1.42886 


26.845 


52 


9 
10 


752 


8.57421 


337 
336 


93o 


9.99969 


754 


8.5745 2 


1.42548 


■637 


5i 


03781 


8-57757 


99929 


9.99969 


03783 


8.57788 


1.42212 


26.432 


50 


ii 


810 


8.58089 


332 


927 


9.99968 


812 


8.58121 


333 


1.41879 


.230 


49 


12 


839 


8.58419 


33° 
328 


926 


9.99968 


842 


8.58451 


33° 
328 
326 
323 


1.41549 


.031 


48 


13 


8b8 


8.58747 


925 


9.99967 


871 


8.58779 


1.41221 


25-835 


47 


14 


897 


8.59072 


325 
323 


924 


9.99967 


900 


8.59105 


1.40895 


.b42 


4b 
45 


15 


0392b 


8-5939$ 


99923 


9.99967 


03929 


8.59428 


1.40572 


25-452 


lb 


955 


8.59715 


320 

•2T8 


922 


9.99966 


958 


8-59749 


321 


1.40251 


.264 


44 


17 


984 


8.60033 


3 IQ 

31b 


921 


9.99966 


987 


8.60068 


3 I 9 
316 


1.39932 


.080 


43 


18 


04013 


8.60349 


919 


9.99965 


0401b 


8.60384 


1. 39616 


24.898 


42 


19 
20 


042 


8.60662 


3 I 3 
3" 


918 


9.99964 


046 


8.60698 


3 X 4 

3ii 
310 

307 
305 
303 
301 

299 
297 
295 
292 

291 
289 
287 
285 
284 
281 


1.39302 


.719 


4i 
40 


04071 


8.60973 


99917 


9.99964 


04075 


8.61009 


1.38991 


24.542 


21 


100 


8.61282 


3°9 


91b 


9.99963 


104 


8.61319 


1.38681 


.368 


39 


22 


129 


8.61589 


3°7 


915 


9.99963 


133 


8.61626 


1.38374 


.19b 


38 


23 


159 


8.61894 


3°5 


913 


9.99962 


ib2 


8.61931 


1.38069 


.02b 


37 


24 
25 


188 


8.62196 


302 
301 

298 
29b 


912 


9.99962 


191 


8.62234 


1.37766 


23-859 


30 


04217 


8.62497 


99911 


9.99961 


04220 


8.62535 


1.37465 


23-695 


35 


2b 


24b 


8.62795 


910 


9.99961 


250 


8.62834 


1.37166 


•532 


34 


27 


275 


8.63091 


909 


9.99960 


279 


8.63131 


1.36869 


.372 


33 


28 


3°4 


8.63385 


294 
293 
290 

288 


907 


9.99960 


308 


8.63426 


1.36574 


.214 


32 


29 

30 


333 


8.63678 


90b 


9-99959 


337 


8.63718 


1.36282 


.058 


3i 


04362 


8.63968 


99905 


9-99959 


0436b 


8.64009 


I.3599I 


22.904 


30 


31 


39i 


8.64256 


287 
284 
283 
281 


904 


9.99958 


395 


8.64298 


1-35702 


.752 


29 


32 


420 


8.64543 


902 


9.99958 


424 


8.64585 


I.354I5 


.602 


28 


33 


449 


8.64827 


901 


9-99957 


454 


8.64870 


I-35I30 


•454 


27 


34 


478 


8.65110 


900 


9.99956 


483 


8.65154 


1.34846 


.308 


2b 


35 


04507 


8.65391 


99898 


9.99956 


04512 


8.65435 


1-3456$ 


22.164 


25 


3° 


536 


8.65670 


279 


897 


9-99955 


54i 


8.65715 


278 
276 
274 

273 
271 
269 
268 


1.34285 


.022 


24 


37 


505 


8.65947 


277 
27b 


896 


9-99955 


570 


8.65993 


1.34007 


21.881 


23 


3« 


594 


8.66223 


894 


9-99954 


599 


8.66269 


I-3373I 


•743 


22 


39 


623 


8.66497 


274 
272 


893 


9-99954 


b28 


8.66543 


1-33457 


.60b 


21 
20 


40 


04653 


8.66769 


99892 


9-99953 


04658 


8.66816 


1-33184 


21.470 


41 


b82 


8.67039 


2b9 
266 


890 


9.99952 


b87 


8.67087 


1. 32913 


•337 


19 


42 


711 


8.67308 


889 


9.99952 


71b 


8.67356 


1.32644 


.205 


18 


43 


740 


8.67575 


888 


9-99951 


745 


8.67624 


2bb 


1.32376 


•075 


17 


44 
45 


769 


8.67841 


263 
263 


886 


9-99951 


774 


8.67890 


264 
263 


1. 321 10 


20.94b 


ib 


04798 


8.68104 


99885 


9.99950 


04803 


8.68154 


1.31846 


20.819 


15 


4 b 


827 


8.68367 


883 


9.99949 


833 


8.68417 


1-31583 


•693 


14 


47 


850 


8.68627 




882 


9-99949 


8b2 


8.68678 




1.31322 


.569 


13 


48 


885 


8.68886 


259 
258 

256 


881 


9.99948 


891 


8.68938 


258 
257 
255 
254 
252 

251 

249 
248 
24b 
245 


1. 31062 


•446 


12 


49 
50 


914 


8.69144 


879 


9.99948 


920 


8.69196 


1.30804 


•325 


11 
TO 


04943 


8.69400 


99878 


9.99947 


04949 


8.69453 


I.30547 


20.206 


.Si 


972 


8.69654 


254 


87b 


9.99946 


978 


8.69708 


1.30292 


.087 


9 


52 


05001 


8.69907 


253 
252 
250 
249 


875 


9.99946 


05007 


8.69962 


1.30038 


19.970 


8 


53 


030 


8.70159 


873 


9-99945 


037 


8.70214 


1.29786 


•855 


7 


54 


059 


8.70409 


872 


9.99944 


06b 


8.70465 


1-29535 


•740 


b 


55 


05088 


8.70658 


99870 


9.99944 


05095 


8.70714 


1.29286 


19^27 


5 


56 


117 


8.70905 


247 
24b 


8b 9 


9-99943 


124 


8.70962 


1.29038 


.51b 


4 


57 


14b 


8.71151 


867 


9.99942 


153 


8.71208 


1.28792 


.405 


3 


5« 


175 


8.71395 


244 


8bb 


9.99942 


182 


8.71453 


1.28547 


.296 


2 


59 


205 


8.71638 


243 


8b4 


9.99941 


212 


8.71697 


243 


1.28303 


.188 


1 


60 


234 


8.71880 


242 


863 


9.99940 


241 


8.71940 


1.28060 


.081 







Nat. COS Log. d. 


Nat. Sin Log. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


t 



87 c 



3° 



t 


Nat. S 


in Log. d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







05234 


8.71880 




99863 


9.99940 


05341 


8.71940 


241 
239 
239 
237 
236 


1.28060 


19.081 


60 


I 


263 


8.72120 


4 


861 


9.99940 


270 


8.72181 


1. 27819 


18.976 


59 


2 


292 


8.72359 


2 39 
238 


860 


9-99939 


299 


8.72420 


1.27580 


.871 


58 


3 


321 


8.72597 


858 


9.99938 


328 


8.72659 


1.27341 


.768 


57 


4 
5 


35o 


8.72834 


2 37 
235 


857 


9.99938 


357 


8.72896 


1.27104 


.666 


56 
55 


05379 


8.73069 


99855 


9-99937 


05387 


8.73132 


1.26868 


18.564 


6 


408 


8.73303 


234 


854 


9.99936 


416 


8.73366 


234 


1.26634 


.464 


54 


7 


437 


8.73535 


232 


8^2 


9.99936 


445 


8.73600 


234 


1.26400 


.366 


53 


8 


466 


8.73767 


232 
230 
229 


851 


9-99935 


474 


8.73832 


232 
231 
229 
229 
227 


1.26168 


.268 


52 


9 
10 


495 


8.73997 


849 


9-99934 


503 


8.74063 


1.25937 


.171 


5i 
50 


05524 


8.74226 


99847 


9-99934 


05533 


8.74292 


1.25708 


18.075 


ii 


553 


8.74454 




846 


9-99933 


5&2 


8.74521 


1.25479 


17.980 


49 


12 


582 


8.74680 


226 


844 


9.99932 


591 


8.74748 


1.25252 


.886 


48 


13 


611 


8.74906 




842 


9.99932 


620 


8-74974 


225 


1.25026 


•793 


47 


14 


640 


8-75 I 30 


224 


841 


9-99931 


649 


8.75199 


1.24801 


.702 


46 


15 


05669 


8-75353 


223 


99839 


9.99930 


05678 


8.75423 7Z 


1-24577 


17.611 


45 


ib 


698 


8-75575 




838 


9.99929 


708 


8.75645 




1-24355 


.521 


44 


i7 


727 


8-75795 




836 


9.99929 


737 


8.75867 




1-24133 


•43i 


43 


18 


756 


8.76015 


219 
217 


834 


9.99928 


766 


8.76087 


219 


1.23913 


•343 


42 


19 


7»5 


8.76234 


833 


9.99927 


795 


8.76306 


1.23694 


.256 


4i 


20 


05814 


8.76451 


99831 


9.99926 


05824 


8.76525 


217 


1-23475 


17.169 


40 


21 


844 


8.76667 


CTfS 


829 


9.99926 


854 


8.76742 


1.23258 


.084 


39 


22 


873 


8.76883 


214 


827 


9.99925 


883 


8.76958 


215 


1.23042 


16.999 


38 


23 


902 


8.77097 


826 


9.99924 


912 


8.77173 


1.22827 


.915 


37 


24 


93i 


8.77310 


213 

212 


824 


9.99923 


941 


8.77387 


214 
213 


1.22613 


.832 


36 
35 


25 


05960 


8.77522 


99822 


9.99923 


05970 


8.77600 


1.22400 


16.750 


26 


989 


8-77733 


2IO 


821 


9.99922 


999 


8.7781 1 




1.22189 


.668 


34 


27 


06018 


8-77943 




819 


9.99921 


06029 


8.78022 




1.21978 


.587 


33 


28 


047 


8.78152 




817 


9.99920 


058 


8.78232 




1.21768 


•5°7 


32 


29 


076 


8.78360 


208 


8i5 


9.99920 


087 


8.78441 


208 


I-2I559 


.428 


31 
30 


30 


06105 


8.78568 


99813 


9.99919 


061 16 


8.78649 


1.2135 1 


16.350 


31 


134 


8.78774 


205 


812 


9.99918 


145 


8.78855 


r 


1.21145 


.272 


29 


32 


163 


8.78979 


810 


9.99917 


*75 


8.79061 


205 


1.20939 


.195 


28 


33 


192 


8.79183 


203 
202 


808 


9.99917 


204 


8.79266 


1.20734 


.119 


27 


34 


221 


8.79386 


806 


9.99916 


233 


8.79470 


203 


1.20530 


•043 


26 


35 


06250 


8.79588 


99804 


9-99915 


06262 


8.79673 


1.20327 


I5-969 


25 


36 


279 


8.79789 


20I 


803 


9.99914 


291 


8.79875 


20I 


1.20125 


.895 


24 


37 


308 


8.79990 


I99 
I99 
197 
197 
I96 
195 


801 


9-99913 


321 


8.80076 




1. 19924 


.821 


23 


3« 


337 


8.80189 


799 


9-999I3 


35o 


8.80277 


199 
I98 
I98 
I96 
I96 


1.19723 


.748 


22 


39 
40 


366 
06395 


8.80388 
8.80585 


797 


9.99912 


379 


8.80476 


1-19524 


.676 


21 


99795 


9.9991 1 


06408 


8.80674 


1.19326 


15-605 


20 


41 


424 


8.80782 


793 


9.99910 


438 


8.80872 


1.19128 


•534 


19 


42 


453 


8.80978 


792 


9.99909 


467 


8.81068 


1.18932 


.464 


18 


43 


482 


8.81173 


790 


9.99909 


496 


8.81264 


1.18736 


•394 


17 


44 


5ii 


8.81367 


I94 

193 
I92 


788 


9.99908 


525 


8.81459 


x 95 
194 


1.18541 


.325 


16 


45 


06540 


8.81560 


99786 


9.99907 


06554 


8.81653 


1-18347 


I5-257 


15 


46 


509 


8.81752 


784 


9.99906 


584 


8.81846 


I 93 


1.18154 


.189 


14 


47 


598 


8.81944 


I92 
I90 
I90 
I89 
188 


782 


9.99905 


613 


8.82038 


192 
192 
190 
190 
189 
188 


1. 17962 


.122 


13 


48 


627 


8.82134 


780 


9.99904 


642 


8.82230 


1.17770 


.056 


12 


49 


656 


8.82324 


778 


9.99904 


671 


8.82420 


1. 17580 


14.990 


11 


50 


06685 


8.82513 


99776 


9.99903 


06700 


8.82610 


1.17390 


14.924 


10 


51 


714 


8.82701 


187 
187 
186 


774 


9.99902 


73o 


8.82799 


1.17201 


.860 


9 


52 


743 


8.82888 


772 


9.99901 


759 


8.82987 


tSS 


1.17013 


•795 


8 


53 


773 


8.83075 


770 


9.99900 


788 


8.83175 ; 


1.16825 


.732 


7 


54 


802 


8.83261 


185 
184 
183 
183 
l8l 


768 


9.99899 


817 


8.83361 


186 

185 

184 
184 

182 


1. 16639 


.669 


6 


55 


06831 


8.83446 


99766 


9.99898 


06847 


8.83547 


1-16453 


14.606 


5 


5& 


860 


8.83630 


764 


9.99898 


876 


8.83732 


1.16268 


•544 


4 


57 


889 


8.83813 


762 


9.99897 


905 


8.83916 


1. 16084 


.482 


3 


5* 


918 


8.83996 


760 


9.99896 


934 


8.84100 


1.15900 


.421 


2 


ft 


947 


8.84177 


l8l 


758 


9.99895 


963 


8.84282 


182 


1.15718 


.361 


1 


976 


8.84358 




75 6 


9.99894 


993 


8.84464 


I.I5536 


.301 







Nat. COS Log. d. 


Nat. S 


in Log. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


t 



m 











L 















t 


Nat. Sin Log. d. 


Nat. COS Log 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







06976 8.84358 


181 


99756 


9.99894 


06993 


8.84464 


182 


I-I5536 


14.301 


60 


I 


07005 8.84539 


179 
179 
178 
177 


754 


9.99893 


07022 


8.84646 




"5354 


.241 


59 


2 


034 8.84718 


752 


9.99892 


051 


8.84826 




i-i5i74 


.182 


58 


3 


063 8.84897 


75o 


9.99891 


080 


8.85006 




1.14994 


.124 


S7 


4 


092 8.85075 


748 


9.99891 


no 


8.85185 


179 
178 


1.14815 


.065 


56 


5 


07121 8.85252 


99746 


9.99890 


07139 


8.85363 


1.14637 


14.008 


55 


b 


150 8.85429 


177 
176 
175 
175 
173 

m 


744 


9.99889 


168 


8.85540 


177 


1. 14460 


i3-95i 


54 


7 


179 8.85605 


742 


9.99888 


197 


8.85717 


177 
176 
176 
174 


1.14283 


.894 


S3 


a 


208 8.85780 


740 


9.99887 


227 


8.85893 


1.14107 


.838 


S2 


9 


237 8.85955 


738 


9.99886 


256 


8.86069 


i-i393i 


.782 


51 


10 


07266 8.86128 


99736 


9.99885 


07285 


8.86243 


I.I3757 


13.727 


50 


II 


295 8.86301 


734 


9.99884 


314 


8.86417 


174 


1-13583 


.672 


49 


12 


324 8.86474 


I 73 
171 
171 
171 
169 
169 
169 
167 
168 
166 


73i 


9.99883 


344 


8.86591 


174 


1-13409 


.617 


48 


13 


353 8.86645 


729 


9.99882 


373 


8.86763 


172 


1-13237 


.563 


47 


14 
15 


382 8.86816 


727 


9.99881 


402 


8.86935 


172 
171 


1-13065 


.510 


46 
45 


074 1 1 8.86987 


99725 


9.99880 


07431 


8.87106 


1.12894 


13-457 


lb 


440 8.87156 


723 


9.99879 


461 


8.87277 


171 


1.12723 


.404 


44 


17 


469 8.87325 


721 


9.99879 


490 


8.87447 


170 
169 
169 
168 
167 
167 
166 
165 
165 
165 
163 
163 
163 
161 
162 
160 
160 
160 

159 
158 
158 


i- 12553 


•352 


43 


18 


498 8.87494 


719 


9.99878 


519 


8.87616 


1.12384 


.300 


42 


19 
20 


527 8.87661 


716 


9.99877 


548 


8.87785 


1.12215 


.248 


4i 
40 


°755 6 8.87829 


99714 


9.99876 


07578 


8.87953 


1. 12047 


I3-I97 


21 


585 8.87995 


166 

165 
164 

164 
163 
163 


712 


9-99875 


607 


8.88120 


1.11880 


.146 


39 


22 


614 8.88161 


71O 


9.99874 


636 


8.88287 


1.11713 


.096 


.38 


23 


643 8.88326 


708 


9.99873 


665 


8.88453 


i-"547 


.046 


37 


24 


672 8.88490 


705 


9.99872 


695 


8.88618 


1.11382 


12.996 


3^ 


25 


07701 8.88654 


99703 


9.99871 


07724 


8.88783 


1.11217 


12.947 


35 


2b 


730 8.88817 


701 


9.99870 


753 


8.88948 


1.11052 


.898 


34 


27 


759 8.88980 


699 


9.99869 


782 


8.89111 


1.10889 


.850 


33 


2b 


788 8.89142 


162 
160 
161 


696 


9.99868 


812 


8.89274 


1. 1 0726 


.801 


32 


29 

30 


817 8.89304 


694 


9.99867 


841 


8.89437 


1-10563 


•754 


3i 
30 


07846 8.89464 


99692 


9.99866 


07870 


8.89598 


1. 1 0402 


12.706 


31 


875 8.89625 


689 


9.99865 


899 


8.89760 


1. 10240 


.659 


29 


32 


904 8.89784 


I 59 


687 


9.99864 


929 


8.89920 


1.10080 


.612 


28 


33 


933 8.89943 


x 59 


685 


9.99863 


958 


8.90080 


1.09920 


.566 


27 


34 
35 


962 8.90102 


J 59 
158 


• 683 


9.99862 


987 


8.90240 


1.09760 


.520 


26 


07991 8.90260 


99680 


9.99861 


08017 


8.90399 


1. 09601 


12.474 


25 


36 


08020 8.90417 


x 57 


678 


9.99860 


046 


8.90557 


1.09443 


.429 


24 


37 


049 8.90574 


J 57 
156 


676 


9.99859 


075 


8.90715 


1.09285 


.384 


23 


3» 


078 8.90730 


673 


9.99858 


104 


8.90872 


I 57 


1.09128 


•339 


22 


39 


107 8.90885 


I 55 
i55 


671 


9-99857 


134 


8.91029 


1 57 
156 


1. 08971 


•295 


21 
20 


40 


08136 8.91040 


99668 


9.99856 


08163 


8.91185 


1.08815 


12.251 


41 


165 8.91 195 


I 55 


666 


9-99855 


192 


8.91340 


I 55 


1.08660 


.207 


19 


42 


194 8.91349 


I 54 


664 


9.99354 


221 


8.91495 


I 55 


1.08505 


.163 


18 


43 


223 8.91502 


I 53 


661 


9-99853 


251 


8.91650 


I 55 


1.08350 


.120 


17 


44 
45 


252 8.91655 


J 53 
152 


659 


9.99852 


280 


8.91803 


*53 
154 


1.08197 


.077 


ib 
T5 


08281 8.91807 


99657 


9.99851 


08309 


8.91957 


1.08043 


12.035 


46 


310 8.91959 


I 5 2 


654 


9.99850 


339 


8.92110 


z 53 


1.07890 


11.992 


14 


47 


339 8.92110 


I 5 I 


652 


9.99848 


368 


8.92262 


I 5 2 


1.07738 


.950 


13 


48 


368 8.92261 


I 5 I 


649 


9.99847 


397 


8.92414 


I 5 2 


1.07586 


.909 


12 


49 
50 


397 8.9241 1 


I 5° 
150 


647 


9.99846 


427 


8.92565 


I 5 I 

151 


1-07435 


.867 


n 


08426 8.92561 


99644 


9.99845 


08456 


8.92716 


1.07284 


11.826 


10 


.Si 


455 8.92710 


149 


642 


9.99844 


485 


8.92866 


I 5° 


1-07134 


.785 


9 


52 


484 8.92859 


149 
148 


639 


9.99843 


514 


8.93016 


I 5° 


1.06984 


•745 


8 


S3 


513 8.93007 


637 


9.99842 


544 


8.93165 


149 
148 
149 


1.06835 


•705 


7 


54 


542 8.93154 


147 
147 


635 


9.99841 


573 


8-93313 


1.06687 


.664 


6 


55 


08571 8.93301 


99632 


9.99840 


08602 


8.93462 


1.06538 


11.625 


5 


S6 


600 8.93448 


147 
146 
146 


630 


9.99839 


632 


8.93609 


147 


1. 06391 


.585 


4 


57 


629 8.93594 


627 


9.99838 


661 


8-93756 


147 


1.06244 


.546 


3 


# 


658 8.93740 


625 


9-99837 


690 


8.93903 


147 
146 
146 


1.06097 


•5°7 


2 


S9 


687 8.93885 


I 45 


622 


9.99836 


720 


8.94049 


I-0595 1 


.468 


1 


60 


716 8.94030 


I 45 


619 


9.99834 


749 


8.94195 


1.05805 


•430 







Nat. COS Log. d. 


Nat. Sin Log. 


Nat. Cot Log. 


c.d. Log. Tan Nat. 


r 



m 













£ 


>° 












f 


Nat. S 


in Log. 


d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







08716 


8.94030 




99619 


999834 


08749 


8.94I95 T „H 


1.05805 


11.430 


60 


I 


745 


8.94174 


144 


617 


9.99833 


778 


8.94340 


X <K5 


1.05660 


•392 


59 


2 


774 


8.94317 


x 43 


614 


9.99832 


807 


8.94485 


I 45 


1-05515 


•354 


S8 


3 


803 


8.94461 


144 


612 


9.99831 


837 


8.94630 


J 45 
143 

144 

143 

142 
142 
142 
141 
140 


1.05370 


.316 


57 


4 


831 


8.94603 


142 

143 


609 


9.99830 


866 


8-94773 


1.05227 


.279 


56 


5 


08860 


8.94746 


99607 


9.99829 


08895 


8.94917 


1.05083 


11.242 


55 


6 


889 


8.94887 


141 


604 


9.99828 


925 


8.95060 


1.04940 


.205 


54 


7 


918 


8.95029 


142 


602 


9.99827 


954 


8.95202 


1.04798 


.168 


53 


8 


947 


8.95170 


141 


599 


9.99825 


983 


8-95344 


1.04656 


.132 


52 


9 
10 


976 


8.95310 


140 
140 


596 
99594 


9.99824 
9.99823 


09013 


8.95486 


1.04514 


•095 


5i 
50 


09005 


8-95450 


09042 


8.95627 


1-04373 


11.059 


ii 


o34 


8.95589 


I 39 


59i 


9.99822 


071 


8.95767 


1.04233 


.024 


49 


12 


063 


8.95728 


I 39 


588 


9.99821 


101 


8.95908 


141 

139 
140 

138 


1.04092 


10.988 


48 


1.3 


092 


8.95867 


I 39 
138 
138 


586 


9.99820 


130 


8.96047 


1-03953 


•953 


47 


14 
15 


121 


8.96005 


583 


9.99819 


159 


8.96l87 


1-03813 


.918 


46 
45 


09150 


8.96143 


99580 


9.99817 


09189 


8.96325 


1.03675 


10.883 


16 


179 


8.96280 


I 37 


-578 


9.99816 


218 


8.96464 


z 39 
138 


1.03536 


.848 


44 


17 


208 


8.96417 


I 37 
136 
136 
136 
135 
135 
134 


575 


9.99815 


247 


8.96602 


1.03398 


.814 


43 


l8 


237 


8.96553 


572 


9.99814 


277 


8.96739 


I 37 
138 
136 
137 
135 
136 


1. 03261 


.780 


42 


19 
20 


266 


8.96689 


570 


9.99813 


306 


8.96877 


1.03123 


.746 


41 
40 


09295 


8.96825 


99567 


9.99812 


09335 


8.97013 


1.02987 


10.712 


21 


324 


8.96960 


564 


9.99810 


365 


8.97150 


1.02850 


.678 


39 


22 


353 


8.97095 


562 


9.99809 


394 


8.97285 


1-02715 


.645 


38 


23 


382 


8.97229 


559 


9.99808 


423 


8.97421 


1.02579 


.612 


V 


24 

25 


411 


8.97363 


x 34 
133 


556 


9.99807 


453 


8-97556 


I 35 

135 


1.02444 


•579 


36 
35 


09440 


8.97496 


99553 


9.99806 


09482 


8.97691 


1.02309 


10.546 


26 


469 


8.97629 


I 33 


55i 


9.99804 


5ii 


8.97825 


I 34 


1-02175 


•5i4 


S4 


27 


498 


8.97762 


133 


548 


9.99803 


54i 


8.97959 


I 34 


1.02041 


.481 


33 


28 


527 


8.97894 


132 


545 


9.99802 


570 


8.98092 


I 33 


1.01908 


•449 


32 


29 
30 


556 


8.98026 


132 
131 


542 


9.99801 


600 


8.98225 


I 33 
*33 


1.01775 


.417 


3i 


09585 


8.98157 


9954° 


9.99800 


09629 


8.98358 


1.01642 


10.385 


30 


31 


614 


8.98288 


I 3 I 


537 


9.99798 


658 


8.98490 


132 


1.01510 


•354 


29 


32 


642 


8.98419 


I 3 I 


534 


9.99797 


688 


8.98622 


132 


1-01378 


.322 


28 


33 


671 


8.98549 


130 


53i 


9.99796 


717 


8.98753 


I 3 I 


1. 01247 


.291 


27 


34 


700 


8.98679 


130 
129 


528 


9-99795 


746 


8.9S884 


I 3 I 
131 


1.01116 


.260 


26 


35 


09729 


8.98808 


99526 


9-99793 


09776 


8.990lg 


1.00985 


10.229 


25 


3& 


758 


8.98937 


129 
129 
128 


523 


9.99792 


805 


8.99145 


130 
130 


1.00855 


.199 


24 


37 


707 


8.99066 


520 


9.99791 


834 


8.99275 


1.00725 


.168 


23 


3» 


816 


8.99194 


128 


517 


9.99790 


864 


8.99405 


130 


1.00595 


.138 


22 


39 


845 


8.99322 


128 
127 


514 


9.99788 


893 


8-99534 


129 
128 
129 
128 


1.00466 


.108 


21 


40 


09874 


8.99450 


995" 


9.99787 


09923 


8.99662 


1.00338 


10.078 


20 


4i 


903 


8-99577 


508 


9.99786 


952 


8.99791 


1.00209 


.048 


19 


42 


932 


8.99704 


127 
126 


506 


9.99785 


981 


8.99919 


1.00081 


.019 


18 


43 


961 


8 99830 


T-?6 


503 


9.99783 


IOOII 


9.00046 


127 
128 
127 


0-99954 


9.9893 


17 


44 
45 


990 

100 1 9 


8.99956 
9.00082 


I~0 
126 
125 

"4 


500 


9.99782 


040 


9.00174 


0.99826 


601 


16 


99497 


9.99781 


ioo6q 


9.00301 


0.99699 


9.9310 


15 


46 


048 


9.00207 


494 


9.99780 


099 


9.00427 


6 


0-99573 


021 


14 


47 


077 


9.00332 


491 


9.99778 


128 


900553 


6 


0.99447 


9-8734 


13 


48 


106 


9.00456 


125 
I23 


488 


9.99777 


158 


9.00679 


6 


0.99321 


448 


12 


49 
50 


135. 


9.00581 


485 


9.99776 


187 


9.00805 


125 


0.99195 


164 


11 
10 


10 1 64 


9.00704 


99482 


9-99775 


102 16 


9.00930 


0.99070 


9.7882 


5i 


192 


9.00828 


I24 


479 


9-99773 


246 


9.01055 


I2 5 


0.98945 


601 


9 


52 


221 


9.00951 


123 
123 


476 


9.99772 


275 


9.0II79 


124 


0.98821 


322 


8 


53 


250 


9.01074 


473 


9.99771 


305 


9.01303 


124 


0.98697 


044 


7 


54 


279 


9.01196 


122 


470, 


9.99769 


334 


9.01427 


123 


0.98573 


9.6768 


6 


55 


10308 


9.01318 


99467 


9.99768 


10363 


9.01550 


0.98450 


9-6493 


5 


5^ 


337 


9.01440 


122 


464 


9.99767 


393 


9.01673 


123 


0.98327 


220 


4 


57 


366 


9.01561 


121 


461 


999765 


422 


9.01796 


123 


0.98204 


9-5949 


3 


58 


395 


9.01682 


121 


458 


9.99764 


452 


9.0I9I8 


122 


0.98082 


679 


2 


59 


424 


9.01803 


121 


455 


999763 


481 


9.02040 


122 


0.97960 


411 


1 


60 


453 


9.01923 


I~0 


452 


9.99761 


5io 


9.02162 


122 


0.97838 


144 







Nat. COS Log. 


d. 


Nat. S 


in Log. 


Nat. Cot Log. 


c.d. ! Log. Tan Nat. 


f 



84 c 



6 ( 



f 


Nat. S 


in Log. 


d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







io453 


9.01923 




99452 


9.99761 


105 10 


9.02162 




0.97838 


9.5144 


60 


I 


482 


9.02043 




449 


9.99760 


540 


9.02283 




0.97717 


9.4878 


S9 


2 


5" 


9.02163 




446 


9-99759 


5 6 9 


9.02404 




0.97596 


614 


58 


3 


54o 


9.02283 


119 
118 
119 

118 


443 


9-99757 


599 


9.02525 




0-97475 


352 


57 


4 


5°9 


9.02402 


44o 


9-99756 


628 


9.02645 


121 
119 


0-97355 


090 


56 
55 


5 


10597 


9.02520 


99437 


9-99755 


10657 


9.02766 


0.97234 


9.3831 


6 


626 


9.02639 


434 


9-99753 


687 


9.02885 


0.97115 


572 


54 


7 


055 


9.02757 


117 

118 


43i 


9-99752 


716 


9.03005 


119 

118 


0.96995 


315 


53 


8 


684 


9.02874 


428 


9-99751 


746 


9.03124 


0.96876 


060 


52 


9 
10 


7i3 


9.02992 


117 
117 
116 


424 


9-99749 


775 


9.03242 


119 
118 


0.96758 


9.2806 


5i 
50 


10742 


9.03109 


99421 


9.99748 


10805 


9.03361 


0.96639 


9.2553 


ii 


771 


9.03226 


418 


9.99747 


834 


9-03479 


118 


0.96521 


302 


49 


12 


800 


9-03342 


116 


415 


9-99745 


863 


9-03597 




0.96403 


052 


48 


13 


829 


9-03458 


116 


412 


9-99744 


893 


9.03714 


117 
118 


0.96286 


9.1803 


47 


14 


858 


9-03574 


116 

"5 
115 
114 

115 
113 
114 
114 
113 


409 


9.99742 


922 


9.03832 


116 


0.96168 


555 


46 


15 


10887 


9.03690 


99406 


9.99741 


10952 


9.03948 


0.96052 


9.1309 


45 


ib 


916 


9.03805 


402 


9.99740 


981 


9.04065 


117 
116 


0-95935 


065 


44 


17 


945 


9.03920 


399 


9.99738 


IIOII 


9.04181 


116 


0.95819 


9.0821 


43 


18 


973 


9.04034 


396 


9-99737 


040 


9.04297 


116 


0.95703 


579 


42 


19 


1 1002 


9.04149 


393 


9-99736 


070 


9.04413 


"5 
"5 
"5 


0-95587 


338 


41 
40 


20 


11031 


9.04262 


99390 


9-99734 


1 1099 


9.04528 


0.95472 


9.0098 


21 


060 


9.04376 


386 


9-99733 


128 


9.04643 


0-95357 


8.9860 


39 


22 


089 


9.04490 


383 


9-99731 


!5 8 


9.04758 


0.95242 


623 


38 


23 


118 


9.04603 


380 


9-99730 


187 


9.04873 


IT 5 


0.95127 


387 


37 


24 


147 


9.04715 


113 


377 


9.99728 


217 


9.04987 


114 
114 


0.95013 


152 


36 
35 


25 


11176 


9.04828 


99374 


9.99727 


1 1 246 


9.05101 


0.94899 


8.8919 


26 


205 


9.04940 




370 


9.99726 


276 


9.05214 


"3 


0.94786 


686 


34 


27 


234 


9.05052 


112 


367 


9.99724 


305 


9.05328 


114 


0.94672 


455 


33 


28 


263 


9.05164 




364 


9-99723 


335 


9.05441 


ZI 3 


0-94559 


225 


32 


29 
30 


, 291 


9-05275 


in 


360 


9.99721 


364 


9-05553 


"3 


0.94447 


8.7996 


3i 
30 


1 1320 


9.05386 


99357 


9.99720 


"394 


9.05666 


0-94334 


8.7769 


31 


349 


9.05497 




354 


9.99718 


423 


9.05778 




0.94222 


542 


29 


32 


378 


9.05607 




35i 


9.99717 


452 


9.05890 




0.941 10 


3*7 


28 


33 


407 


9.05717 


ITO 


347 


9.99716 


482 


9.06002 


TTT 


0.93998 


093 


27 


34 


436 


9.05827 


no 

IO9 
I09 


344 


9.99714 


5" 


9.061 13 


III 


0.93887 


8.6870 


26 
25 


35 


1 1465 


9-05937 


99341 


9-997I3 


"541 


9.06224 


0.93776 


8.6648 


30 


494 


9.06046 


337 


9.9971 1 


570 


9-06335 




0.93665 


427 


24 


37 


523 


9.06155 


334 


9.99710 


600 


9.06445 




0-93555 


208 


23 


38 


552 


9.06264 


IO9 

IO8 


33i 


9.99708 


629 


9.06556 


III 


0.93444 


8.5989 


22 


39 
40 


580 


9.06372 


IO9 

I08 


327 


9.99707 


659 


9.06666 


I09 
I IO 


0-93334 


772 


21 
20 


1 1609 


9.06481 


99324 


9-99705 


11688 


9.06775 


0.93225 


8-5555 


41 


638 


9.06589 




320 


9.99704 


718 


9.06885 




o-93"5 


34o 


19 


42 


667 


9.06696 


IO7 
IO8 


3 J 7 


9.99702 


747 


9.06994 


I09 


0.93006 


126 


18 


43 


696 


9.06804 




314 


9.99701 


777 


9.07103 


IO9 


0.92897 


8.4913 


17 


44 


725 


9.0691 1 


IO7 
I07 

I06 


310 


9.99699 


806 


9.0721 1 


I09 
I08 


0.92789 


701 


16 
15 


45 


"754 


9.07018 


99307 


9.99698 


11836 


9.07320 


0.92680 


8.4490 


46 


783 


9.07124 




303 


9.99696 


865 


9.07428 


IO8 


0.92572 


280 


14 


47 


812 


9.07231 


IO7 
I06 
I05 
IO6 


300 


9.99695 


895 


9.07536 




0.92464 


071 


13 


48 


840 


9-07337 


297 


9.99693 


924 


9.07643 


IO7 
IO8 


0.92357 


8.3863 


12 


49 


869 


9.07442 


293 


9.99692 


954 


9.07751 


IO7 
IO6 


0.92249 


656 


11 
10 


50 


11898 


9.07548 


99290 


9.99690 


11983 


9.07858 


0.92142 


8.345o 


Si 


927 


9-07653 


io 5 


286 


9.99689 


12013 


9.07964 


0.92036 


245 


9 


,S2 


956 


9.07758 


io 5 


283 


9.99687 


042 


9.08071 


IO7 

I06 


0.91929 


041 


8 


.S3 


985 


9.07863 


io 5 


279 


9.99686 


072 


9.08177 


IO6 


0.91823 


8.2838 


7 


54 
55 


12014 


9.07968 


io 5 
104 


276 


9.99684 


101 


9.08283 


IO6 
IO6 


0.91717 


636 


b 


12043 


9.08072 


99272 


9.99683 


12131 


9.08389 


0.91611 


8.2434 


5 


S6 


071 


9.08176 


104 


269 


9.99681 


160 


9.08495 




0.91505 


234 


4 


S7 


100 


9.08280 


104 


265 


9.99680 


190 


9.08600 


io 5 


0.91400 


035 


3 


S8 


129 


9.08383 


103 


262 


9.99678 


219 


9.08705 


io 5 


0.91295 


8.1837 


2 


SQ 


158 


9.08486 


103 


2 S 8 


9.99677 


249 


9.08810 


io 5 


0.91 190 


640 


1 


60 


187 


9.08589 




255 


999675 


278 


9.08914 


104 


0.91086 


443 







Nat. COS Log. 


d. 


Nat. S 


in Log. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


r 



83 c 



7° 



t 


Nat. S 


in Log. 


d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. Log. Cot Nat. 







12187 


9.08589 


103 

103 


99255 


999675 


12278 


9.08914 


105 
104 
104 
103 
104 
103 
103 


0.91086 


8.1443 


60 


I 


216 


9.08692 


251 


9.99674 


308 


9.09019 


0.90981 


248 


59 


2 


245 


9.08795 


248 


9.99672 


338 


9.09123 


0.90877 


054 


58 


3 


274 


9.08897 




244 


9.99670 


367 


9.09227 


0.90773 


8.0860 


57 


4 
5 


302 


9.08999 


102 


240 


9.99669 


397 


9.09330 


0.90670 


667 


56 


1 233 1 


9.09101 


99237 


9.99667 


12426 


9.09434 


0.90566 


8.0476 


55 


6 


360 


9.09202 


102 


233 


9.99666 


456 


9-09537 


0.90463 


285 


54 


7 


389 


9.09304 




230 


9.99664 


485 


9.09640 


0.90360 


095 


53 


8 


418 


9.09405 




226 


9.99663 


515 


9.09742 


103 
102 


0.90258 


7.9906 


52 


9 


447 


9.09506 


100 


222 


9.99661 


544 


9.09845 


0.90155 


718 


5i 


10 


12476 


9.09606 


99219 


9.99659 


12574 


9.09947 


0.90053 


7-9530 


50 


ii 


504 


9.09707 




215 


9.99658 


603 


9.10049 




0.89951 


344 


49 


12 


533 


9.09807 


100 


211 


9.99656 


633 


9.10150 




0.89850 


158 


48 


13 


562 


9.09907 


99 
100 

99 

99 
98 


208 


9-99655 


662 


9.10252 


IOI 


0.89748 


7.8973 


47 


14 


59i 


9.10006 


204 


9.99653 


692 


9.10353 


IOI 


0.89647 


789 


46 


15 


12620 


9.10106 


99200 


9.99651 


12722 


9.10454 


0.89546 


7.8606 


45 


ib 


649 


9.10205 


197 


9.99650 


751 


9-10555 




0.89445 


424 


44 


17 


678 


9.10304 


193 


9.99648 


7 8i 


9.10656 




0.89344 


243* 


43 


18 


706 


9.10402 


189 


9.99647 


810 


9.10756 




0.89244 


062 


42 


19 


735 


9.10501 


99 
98 
98 
98 
98 
97 
97 


186 


9.99645 


840 


9.10856 


IOO 


0.89144 


7.7882 


4i 


20 


12764 


9.10599 


99182 


9.99643 


12869 


9.10956 


0.89044 


7.7704 


40 


21 


793 


9.10697 


17S 


9.99642 


899 


9. 1 1 056 


99 
99 
99 
99 
99 
98 
98 
98 
98 

97 
98 

97 
97 
96 

97 
96 
96 
96 
96 

95 
95 
95 
95 
95 
94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
92 
92 

93 
9i 
92 


0.88944 


525 


39 


22 


822 


9.10795 


175 


9.99640 


929 


9-III55 


0.88845 


348 


38 


23 


851 


9.10893 


171 


9.99638 


958 


9-11254 


0.88746 


171 


37 


24 


880 


9.10990 


167 


9-99637 


988 


9-II353 


0.88647 


7.6996 


36 


25 


12908 


9. 1 1 087 


99163 


9-99635 


13017 


9-11452 


0.88548 


7.6821 


35 


2b 


937 


9.11184 


97 


160 


9-99633 


047 


9.ii55i 


0.88449 


647 


34 


27 


966 


9.11281 


97 
96 

97 
96 

96 

95 
96 

95 

95 
95 


1.S6 


9.99632 


076 


9. 1 1 649 


0.88351 


473 


33 


2d 


995 


9- "377 


1 5 2 


9.99630 


106 


9. 1 1747 


0.88253 


301 


32 


29 


13024 


9.11474 


148 


9.99629 


136 


9.11845 


0.88155 


129 


31 


30 


I3°53 


9.11570 


99144 


9.99627 


13165 


9.11943 


0.88057 


7.5958 


30 


3i 


081 


9. 1 1 666 


141 


9.99625 


195 


9.12040 


0.87960 


787 


29 


32 


no 


9.11761 


137 


9.99624 


224 


9.12138 


0.87862 


618 


28 


33 


139 


9.11857 


133 


9.99622 


254 


9.12235 


0.87765 


449 


27 


34 


168 


9.11952 


129 


9.99620 


284 


9.12332 


0.87668 


281 


26 


35 


I3I97 


9.12047 


99^25 


9.99618 


13313 


9.12428 


0.87572 


7.5II3 


25 


.*> 


226 


9.12142 


122 


9.99617 


343 


9.12525 


0.87475 


7.4947 


24 


37 


254 


9.12236 


94 
95 
94 
94 
93 
94 
93 
93 
93 
93 
93 
92 
92 
92 


118 


9.99615 


372 


9.12621 


0.87379 


781 


23 


3« 


283 


9.12331 


114 


9.99613 


402 


9-12717 


0.87283 


015 


22 


39 


312 


9.12425 


no 


9.99612 


432 


9.12813 


0.87187 


451 


21 
20 


40 


I334I 


9.12519 


99106 


9.99610 


13461 


9.12909 


0.87091 


7.4287 


4i 


370 


9.12612 


102 


9.99608 


491 


9.13004 


0.86996 


124 


19 


42 


399 


9.12706 


098 


9.99607 


521 


9.13099 


0.86901 


7.3962 


18 


43 


427 


9.12799 


094 


9.99605 


550 


9.I3I94 


0.86806 


800 


17 


44 


456 


9.12892 


091 


9.99603 


580 


9.13289 


0.86711 


639 


16 


45 


13485 


9.12985 


99087 


9.99601 


13609 


9-13384 


0.86616 


7-3479 


15 


4 b 


514 


9.13078 


083 


9.99600 


639 


9.13478 


0.86522 


319 


14 


47 


543 


9.13171 


079 


9.99598 


669 


9-13573 


0.86427 


160 


13 


48 


572 


9.13263 


075 


9.99596 


698 


9.13667 


0.86333 


002 


12 


49 


600 


9-13355 


071 


9-99595 


728 


9-13761 


0.86239 


7.2844 


11 


50 


13629 


9.13447 


99067 


9-99593 


13758 


9-I3854 


0.86146 


7.2687 


10 


Si 


658 


9-13539 


92 
9i 
92 
9i 
91 
90 

9i 
90 

9i 
90 


063 


9-99591 


787 


9.13948 


0.86052 


531 


9 


52 


687 


9.13630 


o59 


9.99589 


817 


9.14041 


0.85959 


375 


8 


53 


716 


9.13722 


055 


9.99588 


846 


9.14134 


0.85866 


220 


7 


54 


744 


9.13813 


051 


9.99586 


876 


9.14227 


0-85773 


066 


6 


55 


13773 


9.13904 


99047 


9.99584 


13906 


9.14320 


0.85680 


7.1912 


5 


5° 


802 


9-13994 


043 


9.99582 


935 


9.14412 


0.85588 


759 


4 


57 


831 


9.14085 


o39 


9.99581 


965 


9.14504 


0.85496 


607 


3 


5« 


860 


9-I4I75 


035 


9-99579 


995 


9-14597 


0.85403 


455 


2 


59 


889 


9.14266 


031 


9-99577 


14024 


9.14688 


0.85312 


304 


1 


60 


917 


9-14356 


027 


9-99575 


054 


9.14780 


0.85220 


154 







Nat. COS Log. 


d. 


Nat. S 


in Log. 


Nat. Cot Log. 


c.d. lLog.TanNat. 


f 



82 c 



8 C 



r 


Nat. Sin Log. 


d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







13917 


9-14356 


89 
90 

89 
90 
89 
88 


99027 9.99575 


14054 


9.14780 


92 
91 
91 
91 
91 
91 
90 
91 
90 
90 
89 
90 
89 
90 
89 

89 
88 


0.85220 7. 1 154 


60 


i 


946 


9-14445 


023 9-99574 


084 


9.14872 


0.85128 004 


S9 


2 


975 


9-14535 


OI 9 9-99572 


113 


9.14963 


0-85037 7-0855 


58 


3 


14004 


9.14624 


015 9-99570 


i43 


9-I5054 


0.84946 706 


57 


4 


033 


9.14714 


on 9.99568 


i73 


9-i5i45 


0.84855 558 


50 

55 


5 


1406 1 


9.14803 


99006 9.99566 


14202 


9.15236 


0.84764 7.0410 


6 


090 


9.14891 


89 
89 
88 


002 9.99565 


232 


9-I5327 


0.84673 264 


S4 


7 


119 


9.14980 


98998 9.99563 


262 


9-i54i7 


0.84583 117 


S3 


8 


148 


9.15069 


994 9-9956I 


291 


9.15508 


0.84492 6.9972 


52 


9 


177 


9-I5I57 


88 

88 


99o 9-99559 


321 


9-I5598 


0.84402 827 


51 
50 


10 


14205 


9-I5245 


98986 9.99557 


1435 1 


9.15688 


0.84312 6.9682 


ii 


234 


9-15333 


88 


982 9-99556 


38i 


9-15777 


0.84223 538 


49 


12 


263 


9.15421 


87 
88 


978 9-99554 


410 


9.15867 


0.84133 395 


48 


13 


292 


9.15508 


973 9-99552 


44o 


9-I5956 


0.84044 252 


47 


i4 


320 


9-I5596 


87 
87 
87 
87 
86 


909 9-99550 


47o 


9.16046 


0.83954 IIQ 


46 


15 


14349 


9.15683 


98965 9.99548 


14499 


9-16135 


0.83865 6.8969 


45 


16 


37B 


9.15770 


961 9.99546 


529 


9.16224 


0.83776 828 


44 


17 


407 


9-I5857 


957 9-99545 


559 


9.16312 


89 
88 


0.83688 687 


43 


18 


436 


9-15944 


953 9-99543 


588 


9.16401 


0.83599 548 


42 


19 


464 


9.16030 


86 

87 
86 


948 9.99541 


618 


9.16489 


88 
88 


0.8351 1 408 


41 


20 


14493 


9.16116 


98944 9.99539 


14648 


9.16577 


0.83423 6.8269 


40 


21 


522 


9.16203 


940 9-99537 


678 


9.16665 


88 


0-83335 131 


39 


22 


55i 


9.16289 


85 
86 


936 9-99535 


707 


9-i6753 


88 


0.83247 6.7994 


38 


23 


580 


9.16374 


93i 9-99533 


737 


9.16841 


87 
88 

87 
87 
87 
86 


0.83159 856 


37 


24 


608 


9.16460 


85 
86 


927 9-99532 


767 


9.16928 


0.83072 720 


36 
35 


25 


14637 


9-16545 


98923 9-99530 


14796 


9.17016 


0.82984 6.7584 


26 


666 


9. 1 663 1 


85 
8s 


919 9.99528 


826 


9.17103 


0.82897 448 


34 


27 


695 


9.16716 


9i4 9-995 2 6 


856 


9.17190 


0.82810 313 


33 


28 


723 


9.16801 


85 
84 

85 
84 
84 
84 
84 
83 
84 
83 
83 
83 
83 
83 
82 


910 9.99524 


886 


9.17277 


0.82723 179 


32 


29 
30 


752 


9.16886 


906 9.99522 


915 


9-I7363 


87 
86 


0.82637 045 


3i 


14781 


9.16970 


98902 9.99520 


14945 


9.17450 


0.82550 6.6912 


30 


31 


810 


9-I7055 


897 9-995 J 8 


975 


9-I7536 


86 


0.82464 779 


29 


32 


838 


9-I7I39 


893 9-99517 


15005 


9.17622 


86 


0.82378 646 


28 


33 


867 


9.17223 


889 9.99515 


034 


9.17708 


86 


0.82292 514 


27 


34 


896 


9.17307 


884 9.99513 


064 


9.17794 


86 

Re 


0.82206 383 


26 
25 


35 


14925 


9-I739I 


98880 9.99511 


15094 


9.17880 


0.82120 6.6252 


3^ 


954 


9.17474 


876 9-99509 


124 


9-I7965 86 


0.82035 122 


24 


37 


982 


9-17558 


871 9.99507 


153 


9-18051 °° 


0.81949 6.5992 


23 


3« 


15011 


9.17641 


867 9.99505 


183 


9.18136 


85 

85 

85 
R/i 


0.81864 863 


22 


39 


040 


9.17724 


863 9-99503 


213 


9.18221 


0.81779 734 


21 


40 


15069 


9.17807 


98858 9.99501 


15243 


9.18306 


0.81694 6.5606 


20 


41 


097 


9.17890 


854 9.99499 


272 


9.18391 


0.81609 478 


19 


42 


126 


9-17973 


849 9.99497 


302 


9.18475 


85 
84 

84 
84 
84 
83 
84 
83 
83 
83 
83 
83 
83 
82 


0.81525 350 


18 


43 


155 


9.18055 


82 


845 9.99495 


332 


9.18560 


0.81440 223 


17 


44 


184 


9.18137 


83 
82 


841 9.99494 


362 


9.18644 


0.81356 097 


16 
15 


45 


15212 


9.18220 


98836 9.99492 


I539I 


9.18728 


0.81272 6.4971 


46 


241 


9.18302 


81 


832 9.99490 


421 


9.18812 


0.81188 846 


14 


47 


270 


9.18383 


82 


827 9.99488 


45i 


9.18S96 


0.81104 721 


13 


48 


299 


9.18465 


82 


823 9.99486 


481 


9.18979 


0.81021 596 


12 


49 
50 


327 


9.18547 


81 


818 9.99484 


5" 


9.19063 


0.80937 472 


n 


15356 


9.18628 


98814 9.99482 


i554o 


9.19146 


0.80854 6.4348 


10 


5i 


385 


9.18709 


81 


809 9.99480 


57o 


9.19229 


0.80771 225 


9 


52 


414 


9.18790 


81 


805 9-99478 


600 


9-I93I2 


0.80688 103 


8 


53 


442 


9.18871 


81 


800 9.99476 


630 


9-19395 


0.80605 6.3980 


7. 


54 


471 


9.18952 


81 
80 


796 9-99474 


660 


9.19478 


0.80522 859 


6 


55 


i55oo 


9-I9033 


98791 9-99472 


15689 


9-i956i 


0.80439 6.3737 


5 


5^ 


529 


9.19113 


80 


787 9-99470 


719 


9.19643 


82 


0.80357 617 


4 


57 


557 


9.19193 


80 


782 9.99468 


749 


9-I9725 


82 


0.80275 496 


3 


SB 


586 


9.19273 


80 


778 9.99466 


779 


9.19807 


82 


0.80193 376 


2 


18 


615 


9-19353 


80 


773 9-99464 


809 


9.19889 


82 


0.80111 257 


1 


643 


9-19433 




769 9.99462 


838 


9.19971 




0.80029 138 







Nat. COS Log. 


d. 


Nat. Sin Log. 


Nat. Cot Log. 


C.d. 


Log. Tan Nat. 


t 



81 



t 


Nat. Sin Log. d. 


Nat. COS Log. 


Nat.Tan Log. 


c.d. 


Log. Cot Nat. 







15643 9.19433 


80 


98769 


9.99462 


15838 


9.19971 


82 


0.80029 


6.3138 


60 


I 


672 9.19513 




764 


9.99460 


868 


9.20053 


81 


0.79947 


019 


59 


2 


701 9.19592 


79 
80 

79 

79 

79 


760 


9-99458 


898 


9.20134 


82 


0.79866 


6.2901 


58 


3 


730 9.19672 


755 


9.99456 


928 


9.20216 


81 


0.79784 


783 


57 


4 
5 


758 9-i975i 
15787 9.19830 


75i 


9-99454 


958 


9.20297 


81 
81 


0.79703 


666 


50 
55 


98746 


9.99452 


15988 


9.20373 


0.79622 


6.2549 


b 


816 9.19909 


74i 


9.99450 


1 60 17 


9.20459 


81 


0.79541 


432 


54 


7 


845 9.19988 


79 


737 


9.99448 


047 


9.20540 


81 


0.79460 


316 


53 


8 


873 9.20067 


79 
78 

78 


732 


9.99446 


077 


9.20621 


80 


0-79379 


200 


52 


9 


902 9.20145 


728 


9.99444 


107 


9.20701 


81 
80 


0.79299 


085 


Si 
50 


10 


1593 1 9.20223 


98723 


9.99442 


16137 


9.20782 


0.79218 


6.1970 


ii 


959 9-20302 


79 
78 
78 
77 
78 
78 


7i8 


9.99440 


167 


9.20862 


80 


0.79138 


856 


49 


12 


988 9.20380 


714 


9.99438 


196 


9.20942 


80 


0.79058 


742 


48 


13 


16017 9.20458 


709 


9-99436 


226 


9.21022 


80 


0.78978 


628 


47 


14 


046 9.20535 


704 


9-99434 


256 


9.21102 


80 


0.78898 


515 


4 b 
45 


15 


16074 9.20613 


98700 


9.99432 


16286 


9.21182 


0.78818 


6.1402 


15 


103 9.20691 


695 


9.99429 


316 


9.21261 


80 ' 


290 


44 


17 


132 9.20768 


77 


690 


9.99427 


346 


9-21341 




0.78659 


178 


43 


lb 


160 9.20845 


77 
77 
77 


686 


9.99425 


376 


9.21420 


79 
79 
79 


0.78580 


066 


42 


19 


189 9.20922 


681 


9.99423 


405 


9.21499 


0.78501 


6.0955 


41 
40 


20 


16218 9.20999 


98676 


9.99421 


-6435 


9.21578 


0.78422 


6.0844 


21 


246 9.21076 


77 


671 


9.99419 


465 


9.21657 


79 


0.78343 


734 


39 


22 


275 9-2H53 


77 
76 


667 


9.99417 


495 


9.21736 


79 


0.78264 


624 


38 


2.3 


304 9.21229 


662 


9-994I5 


525 


9.21814 


78 O.78186 


514 


37 


24 


333 921306 


77 
76 

76 
76 
76 


657 


9-994I3 


555 


9.21893 


79 
78 
78 

--,9. 


0.78107 


4°5 


3& 


25 


16361 9.21382 


98652 


9.99411 


16585 


9.21971 


0.78029 


6.0296 


35 


2b 


390 9.21458 


648 


9.99409 


015 


9.22049 


0.77951 


188 


34 


27 


419 9.21534 


643 


9.99407 


b45 


9.22127 'g 
9.22205 i 11 


0.77873 


080 


33 


2b 


447 9.21610 


638 


9.99404 


674 


0-77795 


5-9972 


32 


29 


476 9.21685 


75 
76 


633 
98629 


9.99402 


704 


9.22283 


78 


0.77717 


865 


3i 


30 


16505 9.21761 


9.99400 


16734 


9.22361 


0.77639 


5-9758 


30 


3i 


533 9-21836 


75 

76 


624 


9.99398 


764 


9.22438 


77 

n9. 


0.77562 


051 


29 


32 


562 9.21912 


619 


9.99396 


794 


9.22516 


7° 


0.77484 


545 


28 


33 


591 9.21987 


75 


614 


9-99394 


824 


922593 


77 


0.77407 


439 


27 


34 


620 9.22062 


75 
75 


609 


9.99392 


854 


9.22670 


77 

77 


0-77330 


333 


2b 


35 


16648 9.22137 


98604 


9.99390 


16884 


9.22747 


0.77253 


5.9228 


25 


3^ 


677 9.2221 1 


74 


600 


9.99388 


914 


9.22824 


77 


0.77176 


124 


24 


37 


706 9.22286 


75 


595 


9-99385 


944 


9.22901 


77 
76 


0.77099 


019 


23 


3» 


734 9-22361 


75 


59° 


999383 


974 


9.22977 


0.77023 


5-89I5 


22 


39 


763 9.22435 


74 
74 


585 


9.99381 


17004 


9-23054 


77 


0.76946 


811 
5.8708 


21 
20 


40 


16792 9.22509 


98580 


9-99379 


17033 


9.23130 


' 1 0.76870 

£ 0-76794 


41 


820 9.22583 


74 


575 


9-99377 


053 


9.23206 


605 


19 


42 


849 9.22657 


74 
74 


570 


9-99375 


093 


9.23283 


11 \ 0.76717 
l 6 0.76641 


502 


18 


43 


878 9.22731 


565 


9-99372 


123 


9-23359 


400 


17 


44 
45 


906 9.22805 


74 

73 


5t>i 


9-99370 


153 


9-23435 


70 

75 
76 


0.76565 


298 


ib 


16935 9-22878 


98556 


9.99368 


17183 


9.23510 


0.76490 


5-8197 


15 


4 b 


964 9.22952 


74 
73 
73 
73 
73 
73 
73 


55i 


9.99366 


213 


9.23586 


0.76414 


095 


14 


47 


992 9.23025 


546 


9.99364 


243 


9.23661 


ll 0.76339 

It °-76263 


5-7994 


13 


48 


17021 9.23098 


54i 


9.99362 


273 


9-23737 


894 


12 


49 
50 


050 9.23171 


536 


9-99359 


303 


9.23812 


13 

75 


0.76188 


794 


11 


17078 9.23244 


98531 


9-99357 


17333 


9.23887 


0.761 13 


57694 


10 


5i 


107 9.23317 


526 


9-99355 


363 


9.23962 


l\ 1 0.76038 
'% 0.75963 


594 


9 


52 


136 9-23390 


521 


9-99353 


393 


9.24037 


495 


8 


53 


164 9.23462 


72 
73 
72 
72 
73 


5i6 


9-99351 


423 


9.24112 


75 0.75888 


396 


7 


54 


193 9-23535 


5ii 


9.99348 


453 


9.24186 


74 
75 


0.75814 


297 


b 


55 


17222 9.23607 


98506 


9.99346 


17483 


9.24261 


0-75739 


5-7 J 99 


5 


50 


250 9.23679 


5°i 


9-99344 


5i3 


9-24335 


74 
75 


0.75665 


IOI 


4 


57 


279 9-23752 


496 


9-99342 


543 


9.24410 


0-75590 


004 


3 


58 


308 9.23823 


7 1 


491 


9.99340 


573 


9.24484 


74 


0-75516 


5.6906 


2 


59 


336 9.23895 


72 


486 


9-99337 


603 


9.24558 


74 1 0.75442 


809 


1 


60 


365 9.23967 


72 


481 


9-99335 


633 


9.24632 


74 i O.75368 


713 







Nat. COS Log. d. 


Nat. Sin Log. 


Nat. Cot Log. 


c.d. Log. Tan Nat. 


t 



m 



10 



f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







17365 9.23967 


72 
71 
71 
72 

71 
7i 
71 
70 

71 

70 

71 
70 
70 
70 
70 
70 
70 
69 
70 
69 
69 
69 
69 
69 
69 
69 
6R 


98481 9.99335 




17633 9.24632 




0.75368 5-6713 


60 


I 


393 9-24039 


476 9-99333 




663 9.24706 


74 
73 


0.75294 617 


59 


2 


422 9.241 10 


47i 9-99331 


3 


693 9.24779 


0.75221 521 


58 


3 


451 9.24181 


466 9.99328 


723 924853 


74 


0.75147 425 


57 


4 


479 9- 2 4253 


461 9.99326 


2 


753 9-24926 


73 
74 


0.75074 329 
0.75000 5.6234 


56 
55 


5 


17508 9.24324 


98455 9-99324 


x 77 8 3 9.25000 


b 


537 9-24395 


450 9.99322 


3 


813 9.25073 


73 


0.74927 140 


54! 


7 


565 9.24466 


445 9-993*9 


843 9-25 J 46 


73 
73 


0.74854 045 


53| 


8 


594 9-24536 


440 9.99317 




873 9.25219 


0.74781 5.5951 


52 j 


9 


623 9.24607 


435 9-993I5 
98430 9.99313 


2 
3 


903 9.25292 


73 
73 


0.74708 857 


5i 
50 


10 


1765 1 9.24677 


17933 9-25365 


0.74635 5-5764 


ii 


680 9.24748 


425 9.99310 


963 9-25437 


72 


0.74563 6 7i 


49; 


12 


708 9.24818 


420 9.99308 


n 


993 9-255!0 


73 


0.74490 578 


48! 


13 


737 9.24888 


414 9.99306 




18023 9-25582 


72 


0.74418 485 


47: 


14 


766 9.24958 


409 9.99304 
98404 9.99301 


3 


053 9-25655 


73 
72 
72 


0-74345 393 


46 


15 


17794 9.25028 


18083 9.25727 


0.74273 5-53°* 


45 


lb 


823 9.25098 


399 9.99299 




113 9-25799 


0.74201 209 


441 


17 


852 9.25168 


394 9-99297 


3 


143 9.25871 


72 


0.74129 118 


43 


lb 


880 9.25237 


389 9.99294 


173 9-25943 


72 


0.74057 026 


42 


19 


909 9-25307 


383 9.99292 


2 


203 9.26015 


72 
71 


0.73985 54936 
0.73914 5.4845 


41 : 
40 


20 


17937 9-25376 


98378 9.99290 


18233 9.26086 


21 


966 9.25445 


373 9.99288 


3 


263 9.26158 


72 


0.73842 755 


39 


22 


995 9-255*4 


368 9.99285 


293 9.26229 


7 1 


0.73771 6&5 


38 


23 


18023 9.25583 


362 9.99283 


~ 


323 9.26301 


72 


0-73699 575 


37 


24 

25 


052 9.25652 


357 9-9928I 
98352 9-99278 


3 


353 9-26372 


7 1 
7i 


0.73628 486 
0-73557 5-4397 


3b 
35 


1808 1 9.25721 


18384 9.26443 


2b 


109 9.25790 


347 9-99276 





414 9.26514 


7 1 


0.73486 308 


34^ 


27 


138 9.25858 


69 

68 


341 9.99274 


3 


444 9.2658$ 


70 
71 
7i 


0.73415 219 


33 


28 


166 9.25927 


336 9.99271 


474 9.26655 


0-73345 131 


32 


29 
30 


195 9-25995 
18224 9.26063 


68 
68 


331 9.99269 


2 


504 9.26726 


0.73274 043 


3i 
30 


98325 9-99267 


18534 9.26797 


0.73203 5-3955 


31 


252 9.26131 


68 


320 9.99264 


3 


564 9.26867 


7° 


0-73I33 868 


29 


32 


281 9.26199 


68 


315 9.99262 


"" 


594 9.26937 


79 
7i 
70 

70 
70 
70 
69 
70 
69 
70 
69 
69 
69 
69 
69 

69 
69 
68 


0-73063 781 


28 


33 


309 9.26267 


68 


310 9.99260 


3 
2 

3 


624 9.27008 


0.72992 694 


27 


34 


338 9.26335 


68 
67 

68 


304 9.99257 
98299 9.99255 


654 9.27078 


0.72922 607 


2b 


35 


18367 9.26403 


18684 9-27148 


0.72852 5.3521 


25 


36 


395 9-26470 


294 9.99252 


714 9.27218 


0.72782 435 


24 


37 


424 9-26538 


67 
67 

67 

67 
6 7 
67 
67 
66 
67 
66 


288 9.99250 




745 9.27288 


0.72712 349 


23 


3« 


452 9.26605 


283 9.99248 


3 

2 


775 9-27357 


0.72643 263 


22 


39 
40 


481 9.26672 


277 9-99245 
98272 9.99243 


805 9.27427 


0-72573 178 


21 


18509 9.26739 


18835 9.27496 


O.72504 5.3093 


20 


41 


538 9.26806 


267 9.99241 


3 


865 9.27566 


0.72434 008 


19 


42 


567 9.26873 


261 9.99238 


895 9.27635 


O.72365 5-2924 


18 


43 


595 9-26940 


256 9.99236 


3 
2 


925 9.27704 


O.72296 839 


17 


44 


624 9.27007 


250 9.99233 


955 9-27773 


0.72227 755 


lb 


45 


18652 9.27073 


98245 9.99231 


18986 9.27842 


O.72158 5.2672 


15 


4 b 


681 9.27140 


240 9.99229 


3 


19016 9.2791 1 


0.72089 588 


14 


47 


710 9.27206 


67 
66 


234 9.99226 


046 9.27980 


O.72020 505 


13 


48 


738 9-27273 


229 9.99224 


3 
2 


076 9.28049 


0.71951 422 


12 


49 


767 9.27339 


66 
66 


223 9.99221 


106 9.281 17 


69 

68 


0.71883 339 


11 


50 


18795 9-27405 


98218 9.99219 


19136 9.28186 


0.71814 5.2257 


10 


5i 


824 9.27471 


66 


212 9.99217 


3 


166 9.28254 


69 

68 


0.71746 174 


9 


52 


852 9-27537 


65 
66 


207 9.99214 


197 9.28323 


0.71677 092 


8 


53 


881 9.27602 


201 9.99212 


3 

2 

3 


227 9.28391 


68 


0.71609 on 


7 


54 


910 9.27668 


66 

65 

65 
66 


196 9.99209 


257 9-28459 
19287 9.28527 


68 
68 


0.71541 5.1929 


b 


55 


18938 9.27734 


98190 9.99207 


9.71473 5.1848 


5 


50 


967 9.27799 


185 9.99204 


317 9.28595 


67 

68 


0.71405 767 


4 


57 


995 9.27864 


179 9.99202 


„ 


347 9.28662 


0.71338 686 


3 


5« 


19024 9.27930 


65 
65 


174 9.99200 


3 


378 9.28730 


68 


0.71270 606 


2 


59 


052 9.27995 


168 9.99197 


408 9.28798 


67 


0.71202 526 


1 


60 


081 9.28060 


163 9.99195 




438 9.28865 


0.71135" 446 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


/ 



79 c 



ir 



f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. Log. Cot Nat. 







19081 


9.28060 


65 
65 
64 

65 
65 
64 
64 
6- 


98163 9.99195 


3 


19438 


9.28865 ! 6g 0.71 135 5-1446 


60 


I 


109 


9.28125 


157 9-99I92 


468 


9-28933 | 6v 0.71067 366 
9.29000 1 J 0.71000 286 
9-29067 £ 0.70933 207 


59 


2 


13a 


9.28190 


152 9.99190 


3 


498 


5« 


3 


167 9.28254 


146 9.99187 


5 2 9 


57 


4 
5 


195 


9.28319 


140 9.99185 


3 


559 


9.29134 


67 

At 


O.70866 128 


5b 


19224 


9.28384 


98135 9.99182 


19589 


9.29201 


0-70799 5- io 49 


55 


6 


2=,2 


9.28448 


129 9.99180 


3 


619 


9.29268 ; £' 0.70732 5.O97O 

9-29335 e 7 0.70665 892 
9.29402 A £ 0.70598 814 


54 


7 


28l 


9.28512 


124 9.99177 


649 


53 


8 


309 


9.28577 


64 
64 
64 
64 
63 
64 
64 

63 
63 
64 
63 
63 
63 
63 
63 
6^ 


118 9-99175 


3 
2 

3 


680 


52 


9 
10 


33« 
I9366 


9.28641 


112 9.99172 


710 


9.29468 


67 

66 


0-70532 73D 


5i 


9.28705 


98107 9.99170 


19740 


9-29535 


0.70465 5.0658 


50 


ii 


395 


9.28769 


101 9.99167 


770 


9.29601 


67 0.70399 581 
6 6 1 0.70332 504 


49 


12 


423 


9-28833 


096 9.99165 


3 


801 


9.29668 


48 


13 


452 


9.28896 


090 9.99162 


831 


9-29734 


66 ; 0.70266 427 


47 


14 


481 


9.28960 


084 9.99160 


3 


861 


9.29800 


66 
66 


0.70200 350 


46 


15 


19509 


9.29024 


98079 9-99I57 


19891 


9.29866 


0.70134 5.0273 


45 


ib 


538 


9.29087 


073 9-99I55 


3 


921 


9.29932 


66 


0.70068 197 


44 


17 


566 


9.29150 


067 9.99152 


952 


9.29998 


66 


0.70002 121 


43 


18 


595 


9.29214 


061 9.99150 


3 
2 

3 


982 


9.30064 


66 


0.69936 045 


42 


19 


623 


9.29277 


056 9.99147 


20012 
20042 


9.30130 




0.69870 4.9969 


41 
40 


20 


19652 


9.29340 


98050 9.99145 


9.30195 Jg 0.69805 4.9894 


21 


680 


9.29403 


044 9.99142 


073 


9.30261 6 0.69739 819 
9-30326 | 6 b 0.69674 744 
9-3039I i aa 0.69609 669 


39! 


22 


709 


9.29466 


039 9.99140 


3 


103 


38 


23 


737 


9.29529 


o33 9-99I37 


133 


37 


24 


766 


9.29591 


63 
62 


027 9.99135 


3 


164 9.30457 


0.69543 594 
A j 0.69478 4.9520 


3& 
35 


25 


19794 


9.29654 


98021 9.99132 


20194 


9.30522 


26 


823 


9.29716 


63 


016 9.99130 


3 
3 


224 


9-30587 i Ar °- 6 94i3 446 
9-30652 j £ \ 0.69348 372 


34 


27 


851 


9.29779 


010 9.99127 


254 


33 


28 


880 


9.29841 




004 9.99124 


285 


9.30717 


^ j 0.69283 298 
-> 0.69218 225 

(t 0.69154 4.9152 
°5 0.69089 078 


32 


29 


908 


9.29903 


62 
62 


97998 9.99122 


3 


315 


9.30782 


3i 
30 


30 


19937 


9.29966 


97992 9-991 19 


20345 


9.30846 


31 


9°5 9-30028 


987 9.99117 


3 


376 


9.3091 1 


29 


32 


994 


9.30090 


61 


981 9.991 14 


406 


9-30975 


04 

Ar- 


0.69025 006 


28 


33 


20022 


9-30151 


62 


975 9.991 12 


3 

3 


436 


9-3I040 | £ 


0.68960 4.8933 


27 


34 


051 


9.30213 


62 
61 


969 9.99109 


466 


9.31104 


3 

65 
64 

6a 


0.68896 860 


26 
25 


35 


20079 


9-30275 


97963 9.99106 


20497 


9.31 168 


0.68832 4.8788 


36 


108 


9-30336 


62 


958 9.99104 


*" 


5 2 7 


9-31233 


0.68767 716 


24 


37 


136 


9-30398 


61 


952 9.99101 


3 


557 


9.31297 


0.68703 644 


23 


3« 


105 


9-30459 


60 


946 9.99099 


3 

3 


588 


9.31361 


fi T 0.68639 573 


22 


39 


193 


9.30521 


61 
61 


940 9.99096 


618 


9-3I425 


64 

Ao 


0.68575 501 


21 


40 


20222 


9.30582 


97934 9-99093 


20648 


9.31489 


0.6851 1 4.8430 


41 


250 


9-30643 




928 9.99091 




679 


9-31 ^ 1 64 
9-3i6l6 1 4 

9-3I679 A 


0.68448 359 


19 


42 


279 


9.30704 


61 


922 9.99088 


3 


709 


0.68384 288 


18 


43 


3°7 


9-30765 


61 


916 9.99086 




739 


0.68321 218 


17 


44 


336 9.30826 


61 
60 


910 9.99083 


3 
3 


770 


9-3I743 


63 
64 
Ao 


0.68257 147 


16 


45 


20364 


9.30887 


97905 9.99080 


20800 


9.31806 


0.68194 4.8077 


15 


46 


393 


9-30947 


61 


899 9-99078 


3 


830 


9.31870 


0.68130 007 


14 


47 


421 


9.31008 


60 
61 


893 9.99075 


861 


931933 ; 2 
931996 I £ 


0.68067 4-7937 


13 


48 


450 


9.31068 


887 9.99072 


3 


891 


0.68004 867 


12 


49 


478 


9.31129 


60 
61 
60 
60 


881 9.99070 


3 


921 


9-32059 


63 


0.67941 798 


11 
10 


50 


20507 


9.31189 


97875 9-99067 


209^2 


9.32122 


0.67878 4.7729 


5i 


535 


9.31250 


869 9.99064 


3 


982 


9.32185 £ 1 0.07815 659 


■ 


52 


503 


9.31310 


863 9.99062 


3 
3 
2 


21013 


9.32248 ] * 

9-32311 i £ 


0.67752 59i 


8 


53 


592 


9-3I370 


60 


857 9.99059 


043 


0.67689 522 


7 


54 
55 


620 


9-3I430 


60 


851 9.99056 


073 


9-32373 


63 


0.67627 453 


6 


20649 


9.31490 


97845 9-99054 


21 104 


9-32436 


0.67564 4.7385 


5 


56 


677 


9-3I549 


59 
60 
60 


839 9.99051 


3 


134 


9-32498 i *„ 


0.67502 317 


4 


57 


706 


9.31609 


833 9-99048 


3 


164 9.32561 ! £ 


0.67439 249 


3 


58 


734 


9.31669 


827 9.99046 


" 


195 


9-32623 | , 
9-32685 j f 
9.32747 , D2 


0.67377 181 


2 


fo 


763 


9.31728 


59 
60 


821 9.99043 


3 


225 


O.67315 114 


1 


791 


9.31788 


815 9.99040 


3 


256 


O.67253 046 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat.CotLog.|c.d. 


Log.TanNat. 


/ 



78 c 









12 


D 








t 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







20791 9-3 I 7 8 8 


59 
60 


97815 9.99040 




21256 9.32747 


63 
62 


0.67253 4.7046 


60 


I 


820 9.31847 


809 9.99038 


3 
3 


286 9.32810 


0.67190 4.6979 


59 


2 


848 9.31907 


59 
59 
59 

59 
59 
59 
58 
59 
59 
58 
58 
59 
58 
58 
58 
58 
58 
58 
58 
57 
58 
57 
58 
57 
57 
58 
57 
57 
57 
56 
57 


803 9.99035 


316 9.32872 


61 


0.67128 912 


58 


3 


877 9.31966 


797 9-99032 


347 9-32933 


6~ 


0.67067 845 


57 


4 


905 9.32025 


791 9-99030 


3 
3 


377 9-32995 


62 

62 


0.67005 779 


56 

55 


5 


2 °933 9-3 2o8 4 


97784 9-99027 


21408 9-33057 


0.66943 4-6712 


6 


962 9.32143 


778 9.99024 


438 9-33H9 


61 


0.66881 646 


54 


7 


990 9.32202 


772 9.99022 


3 
3 
3 


4 6 9 9-33 l8 o 


6^ 


0.66820 580 


53 


8 


21019 9.32261 


766 9.99019 


499 9-33242 


61 


0.66758 514 


52 


9 
10 


047 9-323I9 


760 9.99016 


529 9-33303 
21560 9-33365 


62 
61 


0.66697 448 


5i 


21076 9.32378 


97754 9-990I3 


0.66635 4-6382 


50 


ii 


104 9.32437 


748 9.9901 1 


3 
3 
3 
2 

3 
3 
3 


59o 9-33426 


61 


0.66574 3i7 


49 


12 


132 9.32495 


742 9.99008 


621 9.33487 


61 


0.66513 252 


48 


13 


161 9.32553 


735 9-99005 


6 5i 9-33548 


61 


0.66452 187 


47 


14 


189 9.32612 


729 9.99002 


682 9.33609 


61 
61 


0.66391 122 


46 


15 


21218 9.32670 


97723 9-99000 


21712 9.33670 


0.66330 4- 6 o57 


45 


16 


246 9.32728 


717 9-98997 


743 9-33731 


61 


0.66269 4.5993 


44 


17 


27S 9-32786 


711 9-98994 


773 9-33792 


61 


0.66208 928 


43 


18 


303 9.32844 


705 9.98991 


804 9-33853 


60 


0.66147 864 


42 


19 


331 9.32902 


698 9.98989 


3 
3 
3 


834 9-339I3 


61 
60 


0.66087 800 


41 


20 


21360 9.32960 


97692 9.98986 


21864 9.33974 


0.66026 4.5736 


40 


21 


388 9.33018 


686 9.98983 


895 9-34034 


61 


0.65966 673 


39 


22 


4i7 9-33075 


680 9.98980 


925 9.34095 


*r> 


0.65905 609 


38 


23 


445 9-33I33 


6 73 9-98978 


3 
3 
3 


956 9-34155 Z 


0.65845 546 


37 


24 

25 


474 9-33I90 


667 9.98975 


986 9.34215 


6l 
60 


0.65785 483 


3b 
35 


21502 9.33248 


97661 9.98972 


22017 9-34276 


0.65724 4.5420 


26 


S30 9-33305 


655 9.98969 


047 9-34336 


60 


0.65664 357 


34 


27 


559 9-33362 


648 9.98967 


3 
3 
3 
3 


°78 9-34396 


60 


0.65604 294 


33 


28 


587 9.33420 


642 9.98964 


108 9.34456 


60 


0.65544 232 


32 


29 


616 9.33477 


636 9.98961 


139 9-345*6 


60 


0.65484 169 


3i 
30 


30 


21644 9-33534 


97630 9.98958 


22169 9-34576 


0.65424 4.5107 


31 


672 9-33591 


623 9.98955 


200 9.34635 §* 


0.65365 045 


29 


3 2 


701 9.33647 


617 9.98953 


3 
3 
3 
3 
3 


231 9.34695 j 6 


0-65305 4-4983 


28 


33 


729 9-33704 


611 9.98950 


261 9-34755 k~ 


0.65245 922 


27 


34 


758 9-3376"! 


57 
57 
56 


604 9.98947 


292 9.34814 




0.65186 860 


26 


35 


21786 9.33818 


97598 9-98944 


22322 9.34874 


0.65126 4-4799 


25 


36 


814 9-33874 


592 9.98941 


353 9-34933 %% 
383 9-34992 ^ 


O.65067 737 


24 


37 


843 9-33931 


57 
56 
56 
57 
56 
56 
56 
56 
56 
56 


585 9-98938 


O.65008 676 


23 


3« 


871 9-33987 


579 9-98936 


3 
3 
3 
3 

3 


4i4 9-3505I 


6n 


O.64949 615 


22 


39 


899 9-34043 


573 9-98933 


444 9 35 111 


59 
59 
59 
59 
58 
59 
59 
58 
59 
58 
59 
58 
58 
58 
58 
58 
58 
58 
58 
58 
57 


O.64889 555 


21 


40 


21928 9.34100 


97566 9.98930 


22475 9-35*70 


0.64830 4-4494 


20 


41 


956 9.34156 


560 9.98927 


505 9.35229 


O.64771 434 


19 


42 


985 9.34212 


553 9-98924 


536 9.35288 


0.64712 373 


18 


43 


22013 9.34268 


547 9-98921 


567 9-35347 


0.64653 3 J 3 


17 


44 
45 


041 9.34324 


541 9.98919 


3 
3 
3 
3 
3 
3 
3 


597 9-35405 


0.64595 253 


16 


22070 9.34380 


97534 9.98916 


22628 9.35464 


0.64536 4.4194 


15 


46 


098 9.34436 


528 9.98913 


658 9-35523 


0.64477 !34 


14 


47 


126 9.34491 


55 
56 
55 
56 


521 9.98910 


689 9-3558i 


0.64419 075 


13 


48 


155 9-34547 


515 9.98907 


719 9-35640 


0.64360 015 


12 


49 


183 9.34602 


508 9.98904 


75o 9-35698 


0.64302 4.3956 


11 


50 


22212 9.34658 


97502 9.98901 


22781 9.35757 


0.64243 4.3897 


10 


51 


240 9.34713 


55 
56 
55 


496 9.98898 


811 9.35815 


O.64185 838 


9 


52 


268 9.34769 


489 9.98896 


3 


842 9-35873 


0.64127 779 


8 


S3 


297 9-34824 


483 9.98893 


872 9-35931 


0.64069 721 


7 


54 


325 9-34879 


55 
55 
55 
55 
55 


476 9.98890 


3 
3 
3 


903 935989 


0.6401 1 662 


6 

5 


55 


22353 9-34934 


97470 9.98887 


22934 9.36047 


0-63953 4-36o4 


56 


382 9.34989 


463 9.98884 


964 9.36105 


O.63895 546 


4 


57 


410 9.35044 


457 9.98881 




995 9.36163 


0.63837 488 


3 


58 


438 9-35099 


450 9.98878 


3 


23026 9.36221 


0.63779 430 


2 


5Q 


467 9-35 I 54 


55 


444 9-98875 


3 


056 9.36279 


0.63721 372 


1 


60 


495 9-35209 


55 


437 9-98872 


3 


087 9-36336 


O.63664 315 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


/ 



IT 



13° 



t 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







22495 9-35209 


54 


97437 9-98872 


3 


23087 9-36336 


58 
58 

57 
57 
58 

57 


0.63664 4.3315 


60 


I 


523 9-35263 


430 9.98869 


"7 9-36394 


0.63606 257 


59 


2 


552 9-35318 


55 

55 


424 9.98867 




148 9.36452 


0.63548 200 


58 


3 


580 9-35373 


417 9.98864 


3 


179 9-36509 


0.63491 143 


57 


4 


608 9.35427 


54 
54 


411 9.98861 


3 
3 


209 9-36566 


0.63434 086 


50 


5 


22637 9.35481 


97404 9.98858 


23240 9.36624 


0.63376 4-3029 


55 


6 


665 9-35536 


55 
54 
54 


398 9.98855 


3 
3 
3 


271 9.36681 


0.63319 4.2972 


54 


7 


693 9-35590 


391 9.98852 


301 9.36738 *' n 


0.63262 916 


53 


8 


722 9.35644 


384 9.98849 


332 9-36795 




0.63205 859 


52 


9 


75o 9-35698 


54 
54 


378 9.98846 


3 
3 
3 
3 
3 
3 
3 


363 9-36852 


57 
57 
57 
57 
57 
57 
56 
57 
56 


0.63148 803 


5i 
50 


10 


22778 9-35752 


97371 9.98843 


23393 9-36909 


0.63091 4-2747 


ii 


807 9-35 8 °6 


54 

CA 


365 9.98840 


424 9.36966 


0.63034 691 


49 


12 


835 9-3586o 


54 


358 9.98837 


455 937023 


0.62977 635 


48 


1.3 


863 9-35914 


54 


351 9-98834 


485 9-37080 


0.62920 580 


47 


14 
15 


892 9.35968 


54 
54 


345 9-98831 


516 9.37137 


0.62863 524 


46 
45 


22920 9.36022 


97338 9-98828 


23547 9-37 I 93 


0.62807 4.2468 


16 


948 9-36075 


53 

CA 


331 9.98825 


3 


578 9-37250 


0.62750 413 


44 


17 


977 9-36I29 


54 


325 9.98822 


3 


6oS 9-37306 


0.62694 358 


43 


18 


2 3°°5 9-36i82 


53 


318 9.98819 


3 


639 9-37363 


X 0.62637 303 


42 


19 


033 9.36236 


54 
53 


311 9.98816 


3 
3 
3 


6 70 9-374*9 


5° 


0.62581 248 


4i 


20 


23062 9.36289 


97304 9.98813 


23700 9.37476 


57 
56 
56 
56 
56 
56 
56 
56 
56 
56 
55 
56 
56 
55 
55 
56 
55 


0.62524 4.2193 


40 


21 


090 9.36342 


53 


298 9.98810 


73i 9-37532 


0.62468 139 


39 


22 


118 9.36395 


53 


291 9.98807 


3 


762 9.37588 


0.62412 084 


38 


23 


146 9.36449 


54 


284 9.98804 


3 


793 9-37644 


0.62356 030 


37 


2 4 


175 9-36502 


53 
53 


278 9.98801 


3 
3 


823 9-37700 


0.62300 4.1976 


30 


25 


23203 9-36555 


97271 9-98798 


23854 9-37756 


0.62244 4.1922 


35 


26 


231 9.36608 


53 


264 9.98795 


3 


885 9.37812 


0.62188 868 


34 


27 


260 9.36660 


s 


257 9-98792 


3 


916 9.37868 


0.62132 814 


33 


28 


288 9.36713 


5j 


251 9.98789 


3 


946 9.37924 


0.62076 760 


32 


29 


316 9.36766 


53 
53 
52 
53 


244 9.98786 


3 
3 


977 9-37980 


0.62020 706 


3i 


30 


23345 9-36819 


97237 9-98783 


24008 9.38035 


0.61965 4-1653 


30 


31 


373 9-3687I 


230 9.98780 


3 


039 9-38091 


0.61909 600 


29 


32 


401 9.36924 


223 9-98777 


3 


069 9.38147 


0.61853 547 


28 


33 


429 936976 


52 
52 
53 


217 9.98774 


3 


100 9.38202 


0.61798 493 


27 


34 


458 9-37028 


210 9.98771 


3 
3 


131 9-38257 


0.61743 441 


26 


35 


23486 9.37081 


97203 9.98768 


24162 9.38313 


0.61687 4.1388 


25 


36 


514 9.37133 


52 


196 9.98765 


3 


193 9-38368 


0.61632 335 


24 


37 


542 9.37185 


52 
52 


189 9.98762 


3 


223 9-38423 


55 
56 


0.61577 282 


23 


3« 


571 9-37237 


182 9.98759 


3 


254 9-38479 


0.61521 230 


22 


39 


599 9-37289 


52 
52 
52 
52 


176 9.98756 


3 
3 


285 9-38534 


55 
55 
55 
55 


0.61466 178 


21 


40 


23627 9-37341 


97169 9.98753 


24316 9.38589 


0.61411 4.1126 


20 


41 


656 9-37393 


162 9.98750 


3 


347 9-38644 


0.61356 074 


19 


42 


684 9.37445 


155 9-98746 


4 


377 9-38699 


0.61301 022 


18 


43 


712 9-37497 


52 


148 9-98743 


3 


408 9-38754 


55 


0.61246 4.0970 


17 


44 


740 9-37549 


52 
51 


141 9.98740 


3 
3 


439 9-38808 


54 


0.61192 918 


16 


45 


23769 9.37600 


97134 9.98737 


24470 9-38863 


55 
55 


0.61 137 4.0867 


15 


46 


797 9-37652 


52 


127 9.98734 


3 


501 9.38918 


0.61082 815 


14 


47 


825 9-37703 


5 1 


120 9.98731 


3 


532 9-38972 


54 

55 


0.61028 764 


13 


48 


853 9-37755 


52 


113 9.98728 


3 


562 9.39027 


0.60973 713 


12 


49 


882 9.37806 


5 1 
52 
5i 
5i 
5i 
5i 
5i 
5i 


106 9.98725 


3 
3 


593 9-39082 
24624 9.39136 


55 
54 
54 


0.60918 662 


11 
To 


50 


23910 9-37858 


97100 9.98722 


0.60864 4.061 1 


5i 


938 9-37909 


093 9.98719 


3 


655 9-39I90 


0.60810 560 


9 


52 


966 9.37960 


086 9.98715 


4 
3 


686 9.39245 j g 

717 9-39299 2; 


0-60755 509 


8 


S3 


995 9-38oii 


079 9.98712 


0.60701 459 


7 


54 
55 


24023 9.38062 


072 9.98709 


3 
3 


747 9-39353 


3t 

54 


0.60647 4°8 


6 


24051 9.381 13 


97065 9.98706 


24778 9-39407 


0.60593 4-0358 


5 


5^ 


079 9.38164 


058 9.98703 


3 


809 9.39461 | ** 0.60539 308 


4 


57 


108 9.38215 


5 1 


051 9.98700 


3 


840 9-395I5 rT O.6O485 257 


3 


S« 


136 9.38266 


5 1 
5i 
5i 


044 9.98697 


3 


871 9-39569 c! 0.60431 207 


2 


8 


164 9-38317 


037 9.98694 


3 


902 9.39623 «™ 0.60377 *5 8 


1 


192 9.38368 


030 9.98690 


4 


933 939677 1 i 0.60323 108 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. CotLog. c.d. Log.TanNat. 


r 



76 



14 c 



r 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







24192 9.38368 


5° 
5i 
5° 


97030 9.98690 


3 


24933 9-39677 


54 


0.60323 4.0108 


60 


i 


220 9.38418 


023 9.98687 


964 9-39731 


0.60269 058 


59 


2 


249 9-38469 


015 9.98684 


3 


995 9-39785 


54 


0.60215 009 


58 


3 


277 9-385I9 


008 9.98681 


3 


25026 9.39838 


53 


0.60162 3.9959 


S7 


4 


305 9-38570 


5 1 
50 
5° 
5i 
5° 
50 
50 
50 


001 9.98678 


3 

3 


°5 6 9-3989 2 


54 
53 


0.60108 910 


56 
55 


5 


24333 9-38620 


96994 9.98675 


25087 9-39945 


0.60055 3.9861 


6 


362 9.38670 


987 9.98671 


4 
3 
3 
3 
3 
3 


118 9.39999 


54 

53 


0.60001 812 


54 


7 


390 9.38721 


980 9.98668 


149 9.40052 


0.59948 763 


53 


8 


418 9.38771 


973 9.98665 


180 9.40106 


54 


0.59894 7i4 


52 


9 
10 


446 9.38821 


966 9.98662 


211 9.40159 


53 
53 


0.59841 665 


5i 
50 


24474 9-38871 


96959 9.98659 


25242 9.40212 


0.59788 3-9 6l 7 


ii 


5°3 9-389 2 i 


952 9.98656 


273 9.40266 


54 


0-59734 5^8 


49 


12 


531 9-38971 


5° 

5° 
50 
50 


945 9-98652 


4 
3 


304 940319 


53 
53 


0.59681 520 


48 


13 


559 9-39021 


937 9.98649 


335 9-40372 


O.59628 471 


47 


14 


587 9.39071 


930 9.98646 


3 
3 


366 9.40425 


53 
53 


0-59575 423 


4° 
45 


15 


24615 9.39121 


96923 9.98643 


25397 9-40478 


0.59522 3-9375 


16 


644 9.39170 


49 
5o 


916 9.98640 


3 


428 9.40531 


53 


0.59469 327 


44 


17 


672 9.39220 


909 9.98636 


4 


459 9-40584 


53 


0.59416 279 


43 


18 


700 9.39270 


5° 


902 9.98633 


3 


490 9.40636 


52 


0.59364 232 


42 


19 
20 


728 9.39319 


49 
5o 


894 9.98630 


3 
3 


521 9.40689 


53 
53 


0.5931 1 184 


4i 


24756 9-39369 


96887 9.98627 


25552 9.40742 


0.59258 3-9I36 


40 


21 


784 9.39418 


49 


880 9.98623 


4 


583 9-40795 


53 


0.59205 089 


39 


22 


813 9.39467 


49 


873 9.98620 


3 


614 9.40847 


52 


o.59!53 042 


38 


23 


841 9.39517 


5° 


866 9.98617 


3 


645 9.40900 


53 


0.59100 3.8995 


37 


24 
25 


869 9-39566 


49 
49 


858 9.98614 


3 

4 


676 9.40952 


52 
53 


0.59048 947 


30 
35 


24897 9.39615 


96851 9.98610 


25707 9-4 I oog 


0.58995 3- 8 9oo 


26 


925 9.39664 


49 


844 9.98607 


3 


738 9.41057 


52 


0.58943 8 54 


34 


27 


954 9-397I3 


49 


837 9.98604 


3 


769 9.41 1 09 


52 


0.58891 807 


33 


28 


982 9.39762 


49 


829 9.98601 


3 


800 9.41161 


5 2 


0.58839 760 


32 


29 
30 


25010 9.398 1 1 


49 
49 


822 9.98597 


4 
3 


831 9.41214 


53 
52 


0.58786 714 


3i 
30 


25038 9.39860 


96815 9.98594 


25862 9.41266 


0.58734 3-8667 


31 


066 9.39909 


49 


807 9.98591 


3 


893 9.41318 


5 2 


0.58682 621 


29 


32 


o94 9-39958 


49 
48 


800 9.98588 


3 


924 9.41370 


52 


0.58630 575 


28 


33 


122 9.40006 


793 9-98584 


4 


955 9-4I422 


5 2 


0.58578 5 2 8 


27 


34 


151 9.40055 


49 
48 


786 9.98581 


3 
3 


986 9.41474 


52 
52 


0.58526 482 


26 


35 


25179 9.40103 


96778 9.98578 


26017 9.41526 


0.58474 3.8436 


25 


36 


207 9.40152 


49 
48 


77i 9-98574 


4 


048 9.41578 


5 2 


0.58422 391 


24 


37 


235 9.40200 


764 9.98571 


3 


079 9.41629 


5 1 


0.58371 345 


23 


38 


263 9.40249 


49 
48 

49 

48 
48 
48 
48 
48 


756 9.98568 


3 


no 9.41681 


52 
52 
5i 


0.58319 299 


22 


39 
40 


291 9.40297 


749 9-98565 


3 

4 


141 9.41733 


0.58267 254 


21 


25320 9.40346 


96742 9.98561 


26172 9.41784 


0.58216 3.8208 


20 


41 


348 9.40394 


734 9-98558 


3 


203 9.41836 




O.58164 163 


19 


42 


376 9.40442 


727 9-98555 


3 

4 


235 9.41887 


5 1 

52 


0.58113 118 


18 


43 


404 9.40490 


719 9-9855 1 


266 941939 


0.58061 073 


17 


44 


432 9-40538 


712 9.98548 


3 
3 


297 9.41990 


5 1 
5i 


O.58OIO 02» 


16 
15 


45 


25460 9.40586 


96705 9.98545 


26328 9.42041 


0-57959 3-7983 


46 


488 9.40634 


40 
48 
48 
48 

47 
48 
48 


697 9-9854I 


4 


359 9-42093 


5 2 


0.57907 938 


14 


47 


516 9.40682 


690 9-98538 


3 


390 9.42144 


5 1 

5i 


0.57856 893 


13 


48 


545 9-40730 


682 9.98535 


3 


421 9.42195 


0.57805 848 


12 


49 


573 9-40778 


675 9-9853I 


4 
3 


452 9.42246 


5 1 
5i 


0-57754 804 


n 


50 


25601 9.40825 


96667 9.98528 


26483 9.42297 


0.57703 3.7760 


10 


Si 


629 9.40873 


660 9.98525 


3 


515 9.42348 


5 1 


0.57652 715 


9 


52 


657 9.40921 


653 9-985 2 i 


4 


546 9.42399 


5 1 
5i 
5i 
5i 


O.576OI 67I 


8 


S3 


685 9.40968 


48 
47 
48 


645 9-985!8 


3 


577 9-42450 


0.57550 627 


7 


54 
55 


713 9.41016 


638 9-98515 


3 
4 


608 9.42501 


0-57499 583 


b 


25741 9.41063 


96630 9.9851 1 


26639 9.42552 


0.57448 3-7539 


5 


S& 


769 9.41 11 1 


623 9.98508 


3 


670 9.42603 


5 1 


0-57397 495 


4 


57 


798 9.41 158 


47 


615 9.98505 


3 


701 9.42653 


5° 
5i 


0-57347 45i 


3 


58 


826 9.41205 


47 


608 9.98501 


4 


733 9-42704 


0.57296 408 


2 


59 


854 9.41252 


47 
48 


600 9.98498 


3 


764 942755 


5 1 
50 


0.57245 364 


1 


60 


882 9.41300 


593 9-98494 


4 


795 9.42805 


0.57I95 321 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


/ 



75 c 



15 c 



r 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







25882 9.41300 




96593 9.98494 




26795 942805 


5i 


0.57195 3-7321 


60 


i 


910 9.41347 


47 


585 9.98491 


3 


826 9.42856 


0.57144 277 


S9 


2 


938 941394 


47 
47 


578 9.98488 


3 

4 


857 9.42906 


5° 
5i 


0.57094 234 


58 


3 


966 9.41441 


570 9.98484 


888 9-42957 


0.57043 191 


57 


4 
5 


994 941488 


47 
47 
47 

46 


562 9.98481 


3 
4 
3 
3 


920 9.43007 


5° 
50 
5i 

50 


0.56993 J 48 


56 
55 


26022 9.41535 


96555 9-98477 


26951 9-43057 


0.56943 3.7io5 


6 


050 9.41582 


547 9-98474 


982 9.43108 


0.56892 062 


54 


7 


079 9.41628 


540 9.98471 


27013 9.43158 


0.56842 019 


53 


8 


107 9.41675 


47 


532 9.98467 


4 
3 
4 
3 


044 9.43208 


5° 
5° 
5o 


0.56792 3.6976 


S2 


9 
10 


135 9.41722 


47 
46 


524 9.98464 


076 9.43258 


0.56742 933 


51 

50 


26163 9.41768 


96517 9.98460 


27107 9.43308 


0.56692 3.6891 


ii 


191 9.41815 


47 
46 


509 9-98457 


138 9-43358 


5° 

50 
50 


0.56642 848 


49 


12 


219 9.41861 


502 9.98453 


4 
3 

3 

4 
3 


169 9.43408 


0.56592 806 


48 


IS 


247 9.41908 


47 
46 

47 
46 
46 


494 9-98450 


201 9.43458 


0.56542 764 


47 


14 
15 


275 941954 


486 9-98447 


232 943508 


5° 
50 
49 
50 
50 
49 
5° 


0.56492 722 


46 


26303 9.42001 


96479 9.98443 


27263 9.43558 


0.56442 3.6680 


45 


16 


331 9.42047 


471 9.98440 


294 9-43607 


0.56393 6 38 


44 


17 


359 942093 


463 9.98436 


3 

4 
3 


326 9.43657 


0.56343 59D 


43 


18 


387 9.42140 


46 
46 
46 

46 
46 
46 

45 
46 
46 
46 


456 9.98433 


357 9-43707 


0.56293 554 


42 


19 
20 


415 9.42186 


448 9.98429 


388 9.43756 


0.56244 512 


4i 
40 


26443 9.42232 


96440 9.98426 


27419 9.43806 


0.56194 3.6470 


21 


471 9.42278 


433 9-98422 


3 
4 
3 
3 
4 
3 
4 
3 
4 
3 


451 943855 


49 
50 


0.56145 4 2 9 


39 


22 


500 9.42324 


425 9.98419 


482 9.43905 


0.56095 387 


38 


23 


528 9.42370 


417 9.98415 


5i3 9-43954 


49 
50 
49 
49 


0.56046 346 


37 


24 

25 


556 9.42416 


410 9.98412 


545 9-44004 


0.55996 305 


36 


26584 9.42461 


96402 9.98409 


27576 9.44053 


0-55947 3-6264 


35 


26 


612 9.42507 


394 9.98405 


607 9.44102 


0.55898 222 


34 


27 


640 9.42553 


386 9.98402 


638 9.44151 


49 
50 


0.55849 181 


33 


28 


668 9.42599 


379 9-98398 


670 9.44201 


0-55799 140 


^2 


29 
30 


696 9.42644 


4d 
46 


371 9-98395 


701 9.44250 


49 

49 


0-55750 100 


3i 
30 


26724 9.42690 


96363 9.98391 


27732 9-44299 


0-55701 3- 6o 59 


31 


752 9-42735 


45 
46 


355 9-98388 


764 9-44348 


49 


0.55652 018 


29 


32 


780 9.42781 


347 9-98384 


4 
3 


795 9-44397 


49 


0.55603 3.5978 


28 


33 


808 9.42826 


4;> 
46 

45 

45 
46 


340 9.98381 


826 9.44446 


49 


0-55554 937 


27 


34 
35 


836 9.42872 


332 9-98377 


4 
4 
3 
4 


858 9-44495 


49 
49 
48 
49 


0-55505 897 


26 
25 


26864 942917 


96324 9.98373 


27889 9.44544 


0-55456 3-5856 


36 


892 9.42962 


316 9.98370 


921 9.44592 


0.55408 816 


24 


37 


920 9.43008 


308 9.98366 


952 9-44641 


0-55359 776 


2^ 


3« 


948 9.43053 


45 


301 9.98363 


3 


983 9.44690 


49 
48 

49 


0.553™ 736 


22 


39 
40 


976 9.43098 


45 

45 


293 9-98359 


4 
3 


28015 944738 


0.55262 696 


21 

20~ 


27004 9.43143 


96285 9.98356 


28046 9.44787 


0-55213 3-5656 


41 


032 9.43188 


45 


277 9-98352 


4 


077 9.44836 


49 
48 


0.55164 616 


IQ 


42 


060 9.43233 


45 
45 
45 
44 
45 
45 
45 
44 
45 


269 9.98349 


3 


109 9.44884 


0.55116 576 


18 


43 


088 9.43278 


261 9.98345 


4 
3 
4 


140 9-44933 


49 
48 

48 

49 
48 
48 

48 

49 
48 
48 
48 
48 
48 
j.8 


0.55067 536 


17 


44 
45 


116 943323 


253 9-98342 


172 9.44981 


0.55019 497 


16 

15 


27144 943367 


96246 9.98338 


28203 9.45029 


0.54971 3-5457 


46 


172 9.43412 


238 9.98334 


3 


234 9-45078 


0.54922 418 


14 


47 


200 943457 


2 3° 9-9833 1 


266 9.45126 


0.54874 379 


13 


48 


228 9.43502 


222 9.98327 


3 
4 
3 


297 945 J 74 


0.54826 339 


12 


49 
50 


256 943546 


214 998324 


329 9.45222 


0-54778 300 


11 
10 


27284 943591 


96206 9.98320 


28360 9.45271 


0.54729 3-526i 


5* 


312 9.43635 




198 9.98317 


391 945319 


0.54681 222 





52 


340 9.43680 


45 


190 9.98313 


4 


423 945367 


0.54633 183 


8 


53 


368 9.43724 




182 9.98309 


4 
3 
4 
3 


454 9-454I5 


0.54585 144 


7 


54 
55 


396 943769 


44 


174 9.98306 


486 9.45463 


0-54537 i°5 


6 


27424 943813 


96166 9.98302 


28517 9455H 


054489 3-5067 


5 


^6 


452 943857 




158 9.98299 


549 9-45559 


\- O.54441 028 


4 


57 


480 943901 




150 9.98295 


4 


580 9.45606 


Tg ' °-54394 3-4989 
48 i °-54346 95i 
+ 1 0.54298 912 
4 | 0.54250 874 


3 


58 


508 9.43946 


45 


142 9.98291 


4 

I 


612 9.45654 


2 


59 


536 9.43990 




134 9.98288 


643 9.45702 


I 


60 


564 9.44034 




126 9.98284 


4 


675 9-45750 





Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. Log.Tan Nat. 


/ 



74c 











16 


D 








r 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







27564 9.44034 


44 
44 
44 
44 
43 
44 
44 


96126 9.98284 


3 


28675 945750 




0.54250 3.4874 


60 


i 


592 9.44078 


118 9.98281 


706 9.45797 


47 
48 


0.54203 836 


S9 


2 


620 9.44122 


no 9.98277 




738 9.45845 


0.54155 798 


S8 


3 


648 9.44166 


102 9.98273 


4 
3 
4 
4 
3 


769 9.45892 


47 
48 

47 
48 


0.54108 760 


S7 


4 


676 9.44210 


094 9.98270 


801 9.45940 


0.54060 722 


56 

55 


5 


27704 9.44253 


96086 9.98266 


28832 9.45987 


0.54013 3.4684 


6 


731 9.44297 


078 9.98262 


864 9.46035 


0-53965 646 


S4 


7 


759 9-44341 


070 9.98259 


895 9.46082 


47 
48 


0.53918 608 


S3 


8 


787 9.44385 


44 
43 

44 


062 9.98255 


4 


927 9.46130 


0.53870 57o 


S2 


9 
10 


815 9.44428 


054 9.98251 


4 
3 


958 9.46177 


47 
47 


0.53823 533 


51 
50 


27843 9.44472 


96046 9.98248 


28990 9.46224 


0.53776 3-4495 


ii 


871 9.44516 


44 
43 


037 9.98244 


4 
4 


29021 9.46271 


47 
48 


0.53729 458 


49 


12 


899 9-44559 


029 9.98240 


053 9.46319 


0.53681 420 


48 


1.3 


927 9.44602 


43 


021 9.98237 


3 


084 9.46366 


47 


0.53634 383 


47 


14 


955 9-44646 


44 
43 
44 
43 
43 


013 9.98233 


4 
4 
3 


116 9.46413 


47 
47 
47 


0.53587 346 


46 


15 


27983 9.44689 


96005 9.98229 


29147 9.46460 


0.53540 34308 


45 


16 


2801 1 9.44733 


95997 9-98226 


179 9-46507 


0-53493 271 


44 


17 


o39 9-44776 


989 9.98222 


4 


210 9.46554 


47 


0.53446 234 


43 


18 


067 9.44819 


981 9.98218 


4 


242 9.46601 


47 


0.53399 197 


42 


19 


095 9.44862 


43 
43 
43 


972 9.98215 


3 
4 


274 9.46648 


47 
46 

47 
47 


0-5335 2 160 


41 


20 


28123 9.44905 


95964 9.9821 1 


29305 9.46694 


0.53306 34124 


40 


21 


150 9.44948 


956 9.98207 


4 


337 9-4674I 


0.53259 087 


39 


22 


178 9.44992 


44 


948 9.98204 


3 


368 9.46788 


0.53212 050 


38 


23 


206 9.45035 


43 

42 

43 
43 
43 
43 
43 
42 

43 

42 


940 9.98200 


4 


400 9.46835 


47 
46 

47 


0.53165 014 


37 


24 


234 9.45077 


931 9.98196 


4 
4 


432 9.46881 


0.531 19 3.3977 


36 
35 


25 


28262 9.45120 


95923 9-98192 


29463 9.46928 


0.53072 3-3941 


26 


290 9.45163 


915 9.98189 


3 


495 9-46975 


47 
46 

47 
46 

46 

47 
46 
46 
47 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 
46 

45 
46 
46 
46 

45 

46 

45 
46 

45 
45 
46 
45 
45 
46 

45 


0.53025 904 


34 


27 


318 9.45206 


907 9.98185 


4 


526 9.47021 


0.52979 868 


33 


28 


346 9.45249 


898 9.98181 


4 
4 
3 
4 
4 


558 9.47068 


O.52932 832 


32 


29 


374 9-45292 


890 9.98177 


590 9.471 14 
29621 9.47160 


O.52886 796 


3i 
30 


30 


28402 9.45334 


95882 9.98174 


O.52840 3.3759 


31 


429 9-45377 


874 9.98170 


653 9.47207 


0-52793 723 


29 


32 


457 9-454I9 


865 9.98166 


685 9.47253 


0.52747 687 


28 


33 


485 9.45462 


43 


857 9.98162 


4 


716 9.47299 


O.52701 652 


27 


34 
35 


513 9-45504 


42 
43 


849 9.98159 


3 
4 


748 9-47346 


O.52654 616 


26 
25 


28541 945547 


95841 9.98155 


29780 9.47392 


O.52608 3.3580 


36 


569 9.45589 


42 
43 


832 9.98151 


4 


811 9.47438 


O.52562 544 


24 


37 


597 9-45632 


824 9.98147 


4 


843 9-47484 


0.52516 509 


23 


38 


625 9.45674 


42 


816 9.98144 


3 


875 9-47530 


O.52470 473 


22 


39 
40 


652 9.45716 


42 
42 
43 


807 9.98140 


4 
4 


906 9.47576 


O.52424 438 


21 
20 


28680 945758 


95799 9-98I36 


29938 9.47622 


0.52378 3-34Q2 


41 


708 9.45801 


791 9.98132 


4 


970 9.47668 


O.52332 367 


19 


42 


736 9-45843 


42 
42 
42 

42 


782 9.98129 


3 


30001 947714 


O.52286 332 


18 


43 


764 9.45885 


774 9.98125 


4 


°33 9-47760 


0.52240 297 


17 


44 


792 945927 


766 9.98121 


4 
4 


065 9.47806 


0.52194 261 


16 
15 


45 


28820 9.45969 


95757 9-98ii7 


30097 9.47852 


0.52148 3.3226 


46 


847 9.4601 1 


42 
42 
42 
4i 
42 
42 


749 9-98II3 


4 
3 


128 9.47897 


0.52103 191 


14 


47 


875 9.46053 


740 9.981 10 


160 9.47943 


O.52057 156 


13 


48 


903 9.46095 


732 9.98106 


4 


192 9.47989 


0.52011 122 


12 


49 


931 9.46136 


724 9.98102 


4 
4 


224 9.48035 


0.51965 087 


n 
10 


50 


28959 9.46178 


95715 9.98098 


30255 9.48080 


0.51920 3.3052 


Si 


987 9.46220 


707 9.98094 


4 


287 9.48126 


0.51874 017 


9 


52 


29015 9.46262 


42 
4i 


698 9.98090 


4 


319 9.48171 


0.51829 3.2983 


8 


.S3 


042 9.46303 


690 9.98087 


3 


351 9.48217 


o.5 x 783 948 


7 


54 
55 


070 9.46345 


42 
4i 
42 


681 9.98083 


4 
4 
4 


382 9.48262 


0-5I738 914 


6 


29098 9.46386 


95673 9.98079 


30414 9.48307 


0.51693 3.2879 


5 


S6 


126 9.46428 


664 9.98075 


446 9.48353 


0.51647 845 


4 


57 


154 9.46469 


4 1 


656 9.98071 


4 


478 9.48398 


0.51602 811 


3 


S8 


182 9.4651 1 


42 


647 9.98067 


4 


509 9.48443 


0-5!557 777 


2 


S9 


209 9.46552 


4 1 


639 9.98063 


4 
3 


541 9.48489 


0.51511 743 


1 


60 


237 9.46594 


42 


630 9.98060 


573 948534 


0.51466 709 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log. Tan Nat. 


/ 



73 



Nat. Sin Log. d. 



17° 

Nat. COS Log. d. 



Nat.TanLog. 



d. Log. Cot Nat. 



29237 
265 

293 
321 
348 



0.46594 

946635 
9.46676 
9.46717 
9.46758 



29376 
404 
432 
460 
487 



9.46800 
9.46841 
9.46882 
9.46923 
9.46964 



29515 
543 
57i 
599 
626 



9.47005 

947045 
9.47086 
9.47127 
9.47168 



29654 
682 
710 
737 
765 



9.47209 
9.47249 
9.47290 
947330 
947371 



29793 
821 
849 
876 
904 



9474" 
947452 
9.47492 

947533 
947573 



29932 
960 
987 

30015 
043 



9.47613 

947654 
9.47694 

947734 
9.47774 



30071 
098 
126 

154 
182 



9.47814 

947854 
9.47894 

947934 
9.47974 



30209 

237 
265 
292 
320 



9.48014 
9.48054 
9.48094 

948133 
9.48173 



30348 
376 
403 
43i 
459 



9.48213 
9.48252 
9.48292 
948332 
9.48371 



514 
542 
570 

597 



9.4841 1 
9.48450 
9.48490 
9.48529 



30625 

653 
680 
708 

73 6 



9.48607 
9.48647 
9.48686 
9.48725 
9.48764 



30763 
791 
819 
846 
874 
902 



9.48803 
9.48842 
9.48881 
9.48920 
9.48959 
9.48998 



95 6 30 
622 
613 
605 
596 

95588 
579 
57i 
562 

554 



9.98060 
9.98056 
9.98052 
9.98048 
9-98044 
9.98040 
9.98036 
9.98032 
9.98029 
9.98025 



95545 
536 
528 
519 
5" 



9.98021 
9.98017 
9.98013 
9.98009 
9.98005 



95502 
493 
485 
476 
467 



9.98001 
9.97997 

9-97993 
9.97989 
9.97986 



95459 
45o 
441 

433 

424 



9.97982 
9.97978 
9.97974 
9.97970 
9.97966 



95415 
407 

398 
389 
380 



9.97962 
9-97958 
9-97954 
9-97950 
9.97946 



95372 
363 
354 
345 
337 



9.97942 
9-97938 
9-97934 
9-97930 
9-97926 



95328 
319 
310 
301 
293 



9.97922 
9.97918 
9.97914 
9.97910 
9.97906 



95284 

275 

266 

257 
248 



9.97902 
9.97898 
9.97894 
9.97890 
9.97886 



95240 
231 
222 
213 
204 



9.97882 
9.97878 
9.97874 
9.97870 
9.97866 



95195 
186 
177 
168 
159 



9.97861 
997857 
9-97853 
9.97849 

9-97845 



95 J 5o 
142 

133 
124 

"5 
106 



9.97841 
9-97837 
9-97833 
9.97829 
9.97825 
9.97821 



30573 
605 

637 
669 
700 



9-48534 
9-48579 
9.48624 
9.48669 
9.48714 



30732 
764 

796 
828 



9-48759 
9.48804 
9.48849 
9.48894 
9.48939 



30891 
923 
955 
987 

31019 



9.48984 
9.49029 

9-49073 
9.491 18 
9.49163 



3i°5i 

083 

"5 
147 
178 



9.49207 
9.49252 
9.49296 
949341 
949385 



31210 
242 

274 
306 
338 



9.49430 
9.49474 
9495*9 
9-49563 
9.49607 



31370 
402 

434 

466 



9.49652 
9.49696 
9.49740 
9.49784 
9.49828 



3i53o 
562 

594 
626 
658 



9.49872 
9.49916 
9.49960 
9.50004 
9.50048 



31690 
722 

754 
786 
818 



9.50092 
9.50136 
9.50180 
9.50223 
9-50 2 67 



31850 
882 
914 
946 
978 



9-503IJ 
9-50355 
9-50398 
9.50442 

9-50485 



32010 
042 

074 
106 

139 



9-50529 
9-50572 
9.50616 
9-50659 
9-50703 



32171 
203 

235 

267 

299 



9.50746 
9.50789 

9-50833 
9.50876 
9.50919 



32331 
363 
396 
428 
460 
492 



9.50962 
9.51005 
9.51048 
9.51092 

9.5"35 
9.51178 



0.51466 
0.51421 
0.51376 

0.51331 
0.51286 



3.2709 

675 
641 
607 
573 



0.51241 
0.51 196 

o.5"5i 
0.51 106 
0.51061 



3-2539 
506 
472 
438 
405 



0.51016 
0.50971 
0.50927 
0.50882 
0.50837 



3-2371 
338 
305 
272 

238 



0.50793 
0.50748 

0.50704 
0.50659 
0.50615 



3.2205 
172 

139 
106 

073 



0.50570 
0.50526 
0.50481 
0.50437 
0-50393 



3.2041 
008 

3-1975 
943 
910 



O.50348 
0.50304 
0.50260 
0.50216 
0.50172 



3-rf 



845 
813 
780 

748 



0.50128 
0.50084 
0.50040 
0.49996 
0.49952 



3.1716 
684 
652 
620 



0.49908 
0.49864 
0.49820 
0.49777 
0-49733 



3.I556 
524 
492 
460 
429 



0.49689 
0.49645 
0.49602 
049558 
049515 



3. x 397 
366 

334 
303 
271 



0.49471 
0.49428 
0.49384 
0.49341 
0.49297 



3.1240 
209 
178 
146 
ii5 



0.49254 
0.4921 1 
0.49167 
0.49124 
O.49081 



3.1084 

o53 

022 

3.0991 

961 



868 



0.49038 3.0930 

0.48995 

0.48952 

0.48908 

0.48865 

0.48822 



807 
777 



Nat. COS Log. d. Nat. Sin Log. d. Nat.Cot Log. c.d. Log.TanNat. ' 

72° 



18 c 



f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. c.d. 


Log. Cot Nat. 







30902 9.48998 


39 
39 
39 
38 
39 
39 
38 
39 


95106 9.97821 




32492 9.51 178 




0.48822 3.0777 


60 


I 


929 949037 


097 9.97817 


4 
5 


524 9.51221 


43 


0.48779 746 


59 


2 


957 949076 


088 9.97812 


556 9.51264 


43 


0.48736 716 


58 


3 


985 949i 15 


079 9.97808 


4 
4 
4 
4 
4 


588 9.51306 


42 
43 
43 
43 
43 


0.48694 686 


57 


4 
5 


31012 9.49153 


070 9.97804 


621 9.51349 


0.48651 655 


5b 
55 


31040 9.49192 


95061 9.97800 


32653 9.51392 


O.48608 3.0625 


6 


ob8 9.49231 


052 9-97796 


685 9.51435 


048565 595 


54 


7 


095 9.49269 


o43 9-97792 


717 9-5*478 


0.48522 565 


53 


8 


123 949308 


o33 9-97788 


4 


749 9-5I520 


42 


0.48480 535 


52 


9 


151 949347 


39 
38 
39 
38 
38 
39 
38 
38 
39 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 
38 


024 9.97784 


4 

5 


782 9-5I563 


43 

43 
42 
43 
43 
42 

43 
42 
42 
43 
42 
43 
42 
42 
42 
43 
42 
42 
42 


0.48437 505 


5i 
50 


10 


3 1 178 9.49385 


95015 9.97779 


32814 9.51606 


0.48394 3.0475 


ii 


206 9.49424 


006 9.97775 


4 

4 


846 9.51648 


0.48352 445 


49 


12 


233 949462 


94997 9-9777 1 


878 9.51691 


0.48309 415 


48 


13 


2bi 9.49500 


988 9.97767 


4 


911 9-5 I 734 


0.48266 385 


47 


14 
15 


289 949539 


979 9-97763 


4 
4 
5 
4 


943 9-5!776 


0.48224 356 


40 
45 


31316 949577 


94970 9.97759 


32975 9-5!8i9 


0.48181 3.0326 


lb 


344 949615 


961 9.97754 


33°°7 9-5 l8 6i 


0.48139 296 


44 


17 


372 9.49654 


952 9-97750 


040 9.51903 


0.48097 267 


43 


18 


399 9.49692 


943 9-97746 


4 
4 
4 
4 
5 
4 


072 9.51946 


0.48054 237 


42 


19 


427 949730 


933 9-97742 


104 9.51988 


0.48012 208 


4i 
40 


20 


31454 9.49768 


94924 9.97738 


33136 9.52031 


0.47969 3.0178 


21 


482 9.49806 


9i5 9-97734 


169 9.52073 


0.47927 149 


39 


22 


510 9.49844 


906 9.97729 


201 9.52115 


0.47885 120 


38 


23 


537 949882 


897 9.97725 


233 9.52I57 


0.47843 090 


37 


24 


565 949920 


888 9.97721 


4 
4 


266 9.52200 


0.47800 061 


3o 
35 


25 


31593 949958 


94878 9.97717 


33298 9.52242 


047758 3-0032 


2b 


620 9.49996 


869 9-977I3 


4 

5 


330 9.52284 


0.47716 003 


34 


27 


648 9.50034 


860 9.97708 


363 9.52326 


0.47674 2.9974 


33 


28 


675 9-50072 


851 9.97704 


4 

4 
4 
5 


395 9.52368 


42 
42 
42 


0.47632 945 


32 


29 
30 


703 9.501 10 


842 9.97700 


427 9-524IO 


0.47590 916 


3i 
30 


31730 9-50148 


94832 9.97696 


33460 9.52452 


0.47548 2.9887 


31 


758 9.50185 


37 
38 
38 


823 997691 


492 9.52494 


42 
42 
42 


0.47506 858 


29 


32 


78b 9.50223 


814 9.97687 


4 


524 9.52536 


0.47464 829 


28 


33 


813 9.50261 


805 9.97683 


4 


557 9-52578 


0.47422 800 


27 


34 
35 


841 9.50298 


37 
38 
38 
37 
38 


795 9-97679 


4 
5 
4 


589 9.52620 


42 
4i 
42 
42 


0.47380 772 


2b 

25 


3i8b8 9.50336 


94786 9.97674 


33621 9.52661 


0-47339 2-9743 


3^ 


896 9-50374 


777 9-97670 


654 9.52703 


0.47297 714 


24 


37 


923 9.5041 1 


768 9.97666 


4 


686 9.52745 


0.47255 686 


23 


3« 


951 9.50449 


758 9.97662 


4 
5 
4 
4 
4 
5 


718 9-52787 


42 


O.47213 657 


22 


39 
40 


979 9-50486 


37 
37 
38 


749 9-97657 


751 9.52829 


41 
42 
4i 
42 


O.47171 629 


21 
20 


32006 9.50523 


94740 9.97653 


33783 9-52870 


0.47130 2.9600 


41 


o34 9-5056I 


730 9.97649 


816 9.52912 


O.47088 572 


19 


42 


061 9.50598 


37 
37 
38 

37 


721 9.97645 


848 9.52953 


0.47047 544 


18 


43 


089 9.50635 


712 9.97640 


881 9.52995 


O.47005 515 


17 


44 


116 9.50673 


702 9.97636 


4 
4 


913 9-53037 


42 
41 


0.46963 487 


ib 
15 


45 


32144 9.50710 


94693 9.97632 


33945 9-53078 


O.46922 2.9459 


4 b 


171 9-50747 


37 


684 9.97628 


4 

5 


978 9-53120 


42 


O.46880 431 


14 


47 


199 9.50784 


37 
37 


674 9.97623 


34010 9.53161 


4 1 
4i 


O.46839 4°3 


13 


48 


227 9.50821 


665 9.97619 


4 


043 9-53202 


0.46798 375 


12 


49 
50 


254 9-50858 


37 
38 
37 


656 9.97615 


4 
5 
4 


075 9-53244 


42 
4i 
42 
4i 
4i 
4i 
42 
4i 
4i 
4i 
41 


0.46756 347 


11 


32282 9.50896 


94646 9.97610 


34108 9-53285 


0.46715 2.9319 


10 


.Si 


309 9.50933 


637 9.97606 


140 9-53327 


0.46673 291 


9 


52 


337 9-50970 


37 
37 
36 
37 


627 9.97602 


4 

5 


173 9.53368 


0.46632 263 


8 


.S3 


364 9.51007 


618 9-97597 


205 9.53409 


0.46591 235 


7 


54 
55 


392 9.51043 


609 9-97593 


4 
4 

5 


238 9-53450 


0.46550 208 


b 


32419 9.51080 


94599 9-97589 


34270 9-53492 


0.46508 2.9180 


5 


55 


447 9-5III7 


37 


59o 9-97584 


303 9-53533 


0.46467 152 


4 


57 


474 9-5II54 


37 


580 9.97580 


4 


335 9-53574 


0.46426 125 


3 


.SB 


502 9.51 191 


37 
36 


57i 9.97576 


4 
5 


368 9-536I5 


0.46385 097 


2 


59 


529 9.51227 


561 9.97571 


400 9.53656 


0.46344 070 


1 


60 


557 9-51264 


37 


552 9.97567 


4 


433 9-53697 


4 1 


0.46303 042 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.Tan Nat. 


t 



71 











19° 






• 


/ 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







32557 9-5 I2 64 




94552 


9-97567 




34433 9-53697 , T 


0.46303 2.9042 


60 


I 


584 9.51301 


37 


542 


9-97563 


4 
5 


465 9-53738 


41 


0.46262 015 


59 


2 


612 9.51338 


37 
36 

37 
36 
37 
36 
37 
36 
36 
37 
36 
36 
36 
37 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
36 
35 

36 
36 

35 
36 
35 
36 
35 
36 
35 
36 
35 
36 
35 
35 
36 
35 
35 
35 
35 
35 
35 
35 
36 
34 
35 
35 
35 
35 
35 
35 
34 
35 


533 


9-97558 


498 9-53779 


0.46221 2.8987 


5« 


3 


639 9-5I374 


523 


9-97554 


4 

4 

5 


530 9.53820 


4 1 
4i 
41 
4i 
4i 


0.46180 960 


57 


4 
5 


667 9.51411 
32694 9.51447 


514 


9-97550 


563 9-5386i 


0.46139 933 


50 
55 


94504 


9-97545 


34596 9.53902 


0.46098 2.8905 


6 


722 9.51484 


495 


9-97541 


4 
5 


628 9-53943 


0.46057 878 


54 


7 


749 9-5 I 5 2 o 


485 9-97536 


661 9.53984 


0.46016 851 


53 


8 


777 9-5I557 


476 9-97532 


4 


693 9-54025 Tr, 


0-45975 824 


52 


9 


804 9-5I593 


466 9.97528 


4 
5 
4 


726 9.54065 


t v 


0-45935 797 


5i 
50 


10 


32832 9.51629 


94457 


9-97523 


34758 9.54106 


4 1 
41 


0.45894 2.8770 


ii 


859 9.51666 


447 


9-97519 


791 9.54147 


0.45853 743 


49 


12 


887 9.51702 


438 


9-975I5 


4 

5 


824 9.54187 


4° 
4i. 
4i 

40 


0.45813 716 


48 


13 


914 9.51738 


428 


9.97510 


856 9.54228 


0.45772 689 


47 


14 


942 9.51774 


418 


9.97506 


4 
5 
4 
5 


889 9.54269 


0.45731 662 


46 
45 


15 


32969 9.51811 


94409 


9.97501 


34922 9.54309 


0.45691 2.8636 


16 


997 9-5I847 


399 


9.97497 


954 9-54350 | t: 
987 9-54390 J. 

35020 9.54431 | Tn 


0.45650 609 


44 


17 


33024 9.51883 


390 


9.97492 


0.45610 582 


43 


18 


051 9.51919 


380 


9.97488 


4 
4 
5 
4 
5 
4 
5 
4 
4 
5 


0.45569 556 


42 


19 


079 9-5I955 


370 


9.97484 


O52 9.54471 


4i 


045529 5 2 9 


41 


20 


33106 9.51991 


9436i 


9-97479 


35085 9.54512 


0.45488 2.8502 


40 


21 


134 9.52027 


351 


9-97475 


118 9.54552 1 I 

150 9-54593 ; T 
183 9.54633 1 Tn 


0.45448 476 


39 


22 


161 9.52063 


342 


9.97470 


0.45407 449 


38 


2.3 


189 9.52099 


332 


9.97466 


0.45367 423 


37 


24 


216 9.52135 


322 


9.97461 


216 9.54673 


4i 


0.45327 397 
O.45286 2.8370 


30 
35 


25 


33244 9-5 2 i7i 


94313 


9-97457 


35248 9.54714 


2b 


271 9.52207 


303 


9-97453 


281 9-54754 |^o 
314 9-54794 Tt 


045246 344 


34 


27 


298 9.52242 


293 


9.97448 


0.45206 318 


33 


28 


326 9.52278 


284 


9-97444 


4 
5 
4 
5 
4 
5 
4 
5 


346 9-54835 \ Q 

379 9-54875 * 
35412 9.54915 1 AO 
445 9-54955 T 
477 9-54995 \ Q 
5io 9-55035 J 
543 9-55075 ^ 
35576 9.551 15 AO 


0.45165 291 


32 


29 


353 9-523I4 


274 


9-97439 


0.45125 265 


3i 
30 

29 


30 

3i 


33381 9.52350 
408 9.52385 


94264 
254 


9-97435 
9-97430 


0.45085 2.8239 
0.45045 213 


32 


436 9.52421 


245 


9.97426 


0.45005 187 


28 


33 


463 952456 


235 


9.97421 


0.44965 161 


27 


34 


490 9.52492 


225 


9.97417 


0.44925 135 
0.44885 2.8109 


2b 
25 


35 


33518 9.52527 


942IS 


9.97412 


36 


545 9-52563 


206 


9.97408 


4 
5 


608 9.55155 \ 


0.44845 083 


24 


37 


573 9-52598 


196 


9-97403 


641 9.55195 * 
674 9-55 2 35 To 


0.44805 057 


23 


3« 


600 9.52634 


186 


9-97399 


4 
5 
4 
5 
4 
5 


0.44765 032 


22 


39 
40 


627 9.52669 
33655 9-52705 


176 
"94167 


_9_i97394_ 
9-97390 


707 9-55275 


40 


0.44725 006 


21 


3574o 9.55315 


0.44685 2.7980 


20 


41 


682 9.52740 


157 


9-97385 


772 9-55355 To 
805 9-55395 J o Q 
838 9-55434 6 1 


0.4464$ 955 


19 


42 


710 9.52775 


147 


9.97381 


0.44605 929 


18 


43 


737 9-528ii 


137 


9-97376 


0.44566 903 


17 


44 


764 9.52846 


127 


9-97372 


4 
5 


871 9-55474 


,n 


0.44526 878 


16 
15 


45 


33792 9.52881 


941 1 8 


9-97367 


35904 9-555I4 


4° 


0.44486 2.7852 


46 


819 9.52916 


108 


9-97363 


4 

5 

5 


937 9-55554 o Q 0.44446 827 


14 


47 


846 9-5295I 


098 


9-97358 


969 9-55593 % 0.44407 801 


13 


48 


874 9.52986 


088 


9-97353 


36002 9.55633 *„ 0.44367 776 


12 


49 
50 


901 9.53021 
33929 9-53056 


078 


9-97349 


4 

5 
4 
5 


035 9-55673 


39 


0.44327 751 


11 


94068 


9-97344 


36068 9.55712 


0.44288 2.7725 


10 


5i 


956 9-53092 


058 


9-97340 


101 9-55752 1 r: 
134 9-55791 f 


0.44248 700 


9 


52 


983 9.53126 


049 


9-97335 


0.44209 675 


8 


53 


3401 1 9-53i6i 


039 


9-97331 


4 
5 
4 
5 
5 


167 9-5583I f Q 
199 9-55870 | 2 
36232 9.55910 I 


0.44169 650 


7 


54 


038 9.53196 


029 


9.97326 


0.44130 625 


6 


55 


34065 9.53231 


94019 


9.97322 


0.44090 2.7600 


5 


50 


093 9-53266 


009 


9-973I7 


265 9-55949 % ! 0.44051 575 
298 9-55989 * : 0.4401 1 550 


4 


57 


120 9.53301 


93999 


9.97312 


3 


S« 


147 9-53336 


989 9.97308 


4 

5 


331 9.56028 ; & 0.43972 525 
364 9.56067 f \ 0.43933 5oo 


2 


ft 


175 9-53370 


979 


9-97303 


1 


202 9-53405 


969 9.97299 


4 


397 9 56107 * 0.43893 475 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. c.d.|Log.TanNat. 


/ 



70 c 



' Nat. Sin Log. d. 



20° 

Nat. COS Log. d. Nat.TanLog. c.d 



Log. Cot Nat. 



34202 
229 

257 

284 

3ii 



9.53405 
9-53442 
9-53475 
9-53509 
9-53544 



34339 
366 

393 
421 

448 



9.53578 
9-536I3 
9-53647 
9.53682 
9.53716 



34475 
503 
530 

557 
584 



9-53751 
9-53785 
9.538I9 
953854 
9.53888 



34612 

639 
666 
694 
721 



9-53922 
9-53957 
9-5399E 
9-54025 
9-54059 



34748 
775 
803 
830 
857 



964093 
9.54127 
9.54i6i 
9.54I95 
9.54229 



34884 
912 

939 
966 

993 



964263 
9.54297 
9-5433J 
9-54365 
9-54399 



35° 21 
048 

075 
102 
130 



9-54433 
9-54466 
9-54500 
9-54534 
9-54567 



35iS7 
184 
211 

239 
266 



9.54601 

954635 
9.54668 
9.54702 
9-54735 



35293 
320 

347 
375 
402 



9-54769 
9.54802 
9-54836 
9.54869 
9-54903 



35429 
456 
484 
5" 
538 



9-54936 
9.54969 
9-55003 
9-55036 
9-55069 



35565 
592 
619 
647 
674 



9.55102 
9-55I36 
9-55I69 
9.55202 

9.55235 



36701 
728 

755 
782 
810 
837 



9.55268 
9-55301 
9-55334 
9.55367 
9-55400 
9-55433 



93969 
959 
949 
939 
929 



9.97299 
9.97294 
9.97289 
9.97285 
9.97280 



93919 


9.97276 


909 


9.97271 


899 9.97266 


889 9.97262 


879 9-97257 


93869 


9.97252 


859 


9.97248 


849 


997243 


839 


9.97238 


829 


9-97234 


93819 


9.97229 


809 


9.97224 


799 


9.97220 


789 


9.97215 


779 


9.97210 


93769 9.97206 


759 


9.97201 


748' 


9.97196 


738 


9.97192 


728 


9.97187 


93718 9.97182 


708 9.97178 


698 


9.97173 


688 


9.97168 


677 


9.97163 


93667 


9.97I59 


657 9-97I54 


647 


9.97149 


637 


9.97145 


626 


9.97140 


93616 


9-97135 


606 


9.97130 


596 9.97126 


585 


9.97121 


575 


9.971 16 


93565 9-97i 11 


555 


9.97107 


544 


9.97102 


534 


9.97097 


524 


9.97092 


93514 


9.97087 


503 


9.97083 


493 


9.97078 


483 


9-97073 


472 


9.97068 


93462 


9.97063 


452 


9-97059 


441 


9-97054 


43i 


9.97049 


420 9.97044 



93410 

400 

389 

379 
368 
358 



9-97039 
9-97035 
9.97030 
9.97025 
9.97020 
9.97015 



36397 
430 
463 
496 
529 



9.56107 
9.56146 
9.56185 
9.56224 
9.56264 



36562 


9.56303 


595 


9.56342 


628 


956381 


661 


9.56420 


694 9.56459 


36727 9.56498 


760 


9-56537 


793 


9-56576 


826 


9-566I5 


859 9-56654 


36892 9.56693 


925 


9-56732 


958 9.56771 


991 


9.56810 


37024 


9.56849 


37057 


9.56887 


090 


9.56926 


123 


9-56965 


157 


9.57004 


190 9.57042 


37223 


9.57081 


256 9.57120 


289 9.57I58 


322 


967I97 


355 


9.57235 


37388 9.57274 


422 


9-57312 


455 


9-57351 


488 


9.57389 


521 


9.57428 


37554 


9.57466 


588 9.57504 


621 


9-57543 


654 


9-5758I 


687 


9.57619 


37720 


9-57658 


754 


9.57696 


787 9-57734 


820 


9.57772 


853 9-578io 


37887 


9-57849 


920 


9.57887 


953 


9.57925 


986 9.57963 


38020 


9.58001 


38053 9-58039 


086 


9.58077 


120 


9-58115 


153 


9.58I53 


186 


9.58191 



38220 

253 
286 
320 

353 

386 



9.58229 
9.58267 
9-58304 
9.58342 
9.58380 
9.58418 



0.43893 
0.43854 
0.43815 
0.43776 
0.43736 



2-7475 
45o 
425 
400 
376 



0.43697 
0.43658 
0.43619 
0.43580 
0.43541 



2.7351 
326 
302 
277 
253 



0.43502 
0.43463 
0.43424 

0.43385 
0-43346 



2.7228 
204 
179 

155 
130 



0.43307 
0.43268 
0.43229 
0.43190 
043151 



2.7106 
082 
058 

034 

009 



0431 13 
0.43074 
0.43035 
0.42996 
0.42958 



2.6985 
961 
937 
913 



0.42919 
0.42880 
0.42842 
0.42803 
0.42765 



2.6865 
841 
818 
794 
770 



0.42726 
0.42688 
0.42649 
0.4261 1 
0.42572 



2.6746 

723 
699 

675 

652 



042534 
0.42496 

0.42457 
0.42419 
0.42381 



2.6628 
605 
58i 
558 
534 



0.42342 
0.42304 
0.42266 
0.42228 
0.42190 



2.651 1 
488 

464 
441 
418 



0.42151 
0.421 13 
0.42075 
0.42037 
0.41999 



2.6395 
37i 
348 
325 
302 



0.41961 
0.41923 
0.41885 
0.41847 
0.41809 



2.6279 
256 

233 
210 
187 



0.41771 

0.41733 
0.41696 
0.41658 
0.41620 
0.41582 



2.6165 
142 
119 
096 
074 
051 



Nat. COS Log. d. 



Nat. Sin Log. d. Nat.CotLog. c.d. Log.TanNat. ' 

69° 



2r 



■ 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







35837 9-55433 


33 
33 
33 


93358 9-970I5 


5 
5 


38386 9.58418 


37 
38 
38 
38 
37 
38 
37 
38 
38 
37 
38 
37 

q8 


0.41582 2.6051 


60 


I 


864 9-55466 


348 9.97010 


420 9.58455 


0.41545 028 


59 


2 


891 9-55499 


337 9-97005 


453 9-58493 


0.41507 006 


58 


3 


918 9-55532 


327 9.97001 


4 
5 
5 
5 
5 
5 
5 
5 


487 9.58531 


0.41469 2.5983 


57 


4 


945 955564 


32 
33 
33 


316 9.96996 


520 9.58569 


0.41431 961 


5o 


5 


35973 9-55597 


93306 9.96991 


38553 9-58606 


0.41394 2.5938 


55 


6 


3booo 9.55630 


295 9.96986 


587 9.58644 


0.41356 916 


54 


7 


027 9.55663 


33 
32 
33 
33 


285 9.96981 


620 9.58681 


0.41319 893 


53 


8 


o54 9-55695 


274 9-96976 


654 9.58719 


0.41281 871 


52 


9 


081 9.55728 


264 9.96971 


687 9.58757 


0.41243 848 
0.41206 2.5826 


5i 
50 


10 


36108 9.55761 


93253 9-96966 


38721 9.58794 


ii 


135 9-55793 


32 
33 


243 9.96962 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


754 9.58832 


0.41168 804 


49 


12 


162 9.55826 


232 996957 


787 9.58869 


0.41131 782 


48 


13 


190 9-55858 


32 


222 9.96952 


821 9.58907 


i 0.41093 759 


47 


14 


217 9.55891 


33 
32 
33 
32 
33 
32 
32 


211 9.96947 


854 9.58944 


37 


0.41056 737 


4 b 
45 


15 


36244 9.55923 


93201 9.96942 


38888 9.58981 


0.41019 2.5715 


ib 


271 9-55956 


190 9.96937 


921 9.59019 D 
955 9.59056 % 
988 9.59094 % n 


0.40981 693 


44 


17 


298 9-55988 


180 9.96932 


0.40944 671 


43 


18 


325 9.56021 


169 9.96927 


0.40906 649 


42 


19 
20 


352 9-56053 
36379 9-56085 


159 9.96922 


39022 9.59131 


37 


0.40869 627 


4i 
40 


93148 9.96917 


39055 9.59I68 


0.40832 2.5605 


21 


406 9.561 18 


33 
32 


137 9.96912 


089 9-59205 [ 2r 


0.40795 583 


39 


22 


434 9-56I50 


127 9.96907 


122 9.59243 


37 
37 


0.40757 56i 


38 


2S 


461 9.56182 


32 


116 9.96903 


4 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 

5 
6 


156 9.59280 


0.40720 539 


37 


24 


488 9.56215 


33 
32 
32 
32 
32 
32 
33 


106 9.96898 


190 9-593I7 


0.40683 517 


36 
35 


25 


36515 9.56247 


93°95 9-96893 


39223 9.59354 


0.40646 2.5495 


2b 


542 9.56279 


084 9.96888 


257 9-59391 *' 

290 9-59429 i 7 
324 9.59466 %' n 


0.40609 473 


34 


27 


569 9-563II 


074 9.96883 


0.40571 452 


33 


28 


596 9.56343 


063 9.96878 


0.40534 43o 


32 


29 


623 9-56375 


052 9.96873 


357 9-59503 


37 
37 
37 
37 
37 
37 


0.40497 408 


3i 


30 


36650 9.56408 


93042 9.96868 


39391 9-59540 


0.40460 2.5386 


30 


3i 


677 9.56440 


32 


031 9.96863 


4 2 5 9-59577 


0.40423 365 


29 


32 


704 9.56472 


3 2 
32 


020 9.96858 


458 9-596I4 


0.40386 343 


28 


33 


731 9-56504 


010 9.96853 


49 2 9-5965I 


0.40349 322 


27 


34 
35 


758 9-56536 


32 
32 


92999 9.96848 


526 9.59688 


0.40312 300 


2b 


36785 9-56568 


92988 9.96843 


39559 9-59725 


0.40275 2.5279 


25 


36 


812 9.56599 


3 1 


978 9.96838 


593 9.59762 %, 


0.40238 257 


24 


37 


839 9-5663I 


3 2 


967 9-96833 


626 9-59799 


3/ 
36 

37 
37 
37 
37 
36 


0.40201 236 


2^ 


38 


867 9.56663 


3 2 


956 9.96828 


660 9.59835 


0.40165 214 


22 


39 


894 9-56695 


3 2 
32 


945 9-96823 


694 9-59872 


0.40128 193 


21 


40 


36921 9.56727 


92935 9.96818 


397 2 7 9-59909 


0.40091 2.5172 


20 


41 


948 9-56759 


32 

31 


924 9.96813 


761 9.59946 


0.40054 150 


19 


42 


975 9-56790 


913 9.96808 


795 9-59983 


0.40017 129 


18 


43 


37002 9.56822 


32 


902 9.96803 


829 9.60019 


0.39981 108 


17 


44 


029 9.56854 


3 2 
32 
31 


892 9.96798 


862 9.60056 


37 
37 
37 
36 
37 
37 
36 
37 
36 
37 
36 
37 
36 
37 
36 
37 
36 


0.39944 086 
0.39907 2.5065 


16 
15 


45 


37056 9.56886 


92881 9.96793 


39896 9.60093 


4 b 


083 9.56917 


870 9.96788 


930 9.60130 


0.39870 044 


14 


47 


no 9.56949 


3 2 


859 9.96783 


963 9.60166 


0.39834 023 


13 


48 


137 9-5698o 


3 1 


849 9.96778 


997 9.60203 


0.39797 002 


12 


49 
50 


164 9.57012 


3 2 
32 


838 9.96772 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


40031 9.60240 


0.39760 2.4981 


n 


37i9i 9-57044 


92827 9.96767 


40065 9.60276 


0.39724 2.4960 


10 


5i 


218 9.57075 


3 1 


816 9.96762 


098 9.60313 


0.39687 939 


9 


52 


245 9-57I07 


3 2 


805 9.96757 


132 9.60349 


0.39651 918 


8 


S3 


272 9-57I38 


3 1 


794 9-96752 


166 9.60386 


0.39614 897 


7 


54 


299 9-57 J 69 


3 1 
32 


784 9.96747 


200 9.60422 


0.39578 876 


b 


55 


37326 9.57201 


92773 9.96742 


40234 9.60459 


0.39541 2.4855 


5 


& 


353 9-57232 


3 1 


762 9-96737 


267 9.60495 


0.39505 834 


4 


57 


380 9.57264 


32 
3i 


75i 9-96732 


301 9.60532 


0.39468 813 


3 


58 


407 9.57295 


740 9.96727- 


335 9-60568 


0.39432 792 


2 


•I 9 


434 9-57326 


3 1 


729 9.96722 


369 9.60605 


0-39395 772 


1 


60 


461 9-57358 


32 


718 9.96717 


403 9.60641 


0-39359 .751 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log.|c.d. Log.Tan Nat. 


f 










66 


1° 









22° 



Nat. Sin Log. d. Nat. CoSLog. d. iNat.TanLog. c.d.JLog.CotNat 



3746i 
488 
515 

542 
S69 



9-57358 
9-57389 
9.57420 

9-57451 
9.57482 



37595 
622 
649 
676 
703 



9-57514 
9-57545 
9-57576 
9.57607 
9-57638 



3773o 
757 
784 
811 



9.57669 
9.57700 

9-57731 
9.57762 

9-57793 



3786s 
892 
919 
946 
973 



9.57824 
9-57855 q 
9.57885 f T 
9.57916 

9-57947 



37999 



o53 
080 
107 



9-57978 
9.58008 
9.58039 
9.58070 
9-58ioi 



38134 
161 

188 

215 
241 



9-58i3i 
9.58162 
9.58192 
9.58223 
9-58253 



38268 

295 
322 

349 
376 



9.58284 
9-583I4 
9-58345 
958375 
9.58406 



38403 
43° 
456 
483 
5io 



9-58436 
9.58467 
9-58497 
9-585 2 7 
9-58557 



38537 
564 
59i 
617 
644 



9.58588 
9.58618 
9.58648 
9.58678 
9.58709 



38671 
698 
725 
75 2 
778 



958739 
9.58769 

9-58799 
9.58829 

9-58859 



38805 
832 
859 
886 
912 



9.58889 
9.58919 
9.58949 

9-58979 
9.59009 



38939 
966 

993 

39020 

046 

073 



9-59039 
9.59069 
9.59098 
9.59128 
9.59158 
9.59188 



3i 

I 31 
31 

30 
3i 
31 

3i 
30 
3i 
30 
3i 
30 
3i 
30 
3i 
30 
3i 
30 
3i 
30 
30 
30 
3i 
30 
30 
30 
3i 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
30 
29 
30 
30 
30 



92718 
707 

697 
686 

675 



9.96717 
9.9671 1 
9.96706 
9.96701 
9.96696 



92664 

653 

642 

631 



9.96691 
9.96686 
9.96681 
9.96676 
9.96670 



598 
587 
576 
565 



9.96665 
9.96660 
9.96655 
9.96650 
9.96645 



92554 
543 
532 
521 
5io 



9.96640 
9.96634 
9.96629 
9.96624 
9.96619 



92499 
488 

477 
466 

455 



9.96614 
9.96608 
9.96603 
9.96598 
9-96593 



92444 
432 
421 
410 
399 



9.96588 
9.96582 

9-96577 
9.96572 
9.96567 



377 
366 
355 
343 



9.96562 
9-96556 
9-9655I 
9.96546 
9.96541 



92332 
321 
310 

299 
287 



9.96535 
996530 
9-96525 
9.96520 
996514 



92276 
265 
254 
243 
231 



9.96509 
9.96504 
9.96498 
9.96493 
9.96488 



92220 
209 
198 
186 
175 



9.96483 
9.96477 
9.96472 

9.96467 
9.96461 



92164 
152 
141 
130 
119 



9.96456 
9.96451 
9.96445 
9.96440 
9.96435 



92107 
096 
085 

073 
062 

050 



9.96429 
9.96424 
9.96419 
9.96413 
9.96408 
9.96403 



40403 
436 
470 
504 
538 



9.60641 
9.60677 
9.60714 
9.60750 
9.60786 



40572 
606 
640 
674 
707 



9.60823 
9.60859 
9.60895 
9.60931 
9.60967 



40741 

775 

809 

843 
877 



9.61004 
9.61040 
9.61076 
9.61112 
9.61 148 



409 1 1 
945 
979 

41013 
047 



9.61 184 
9.61220 
9.61256 
9.61292 
9.61328 



4108 1 

ii5 
149 

183 
217 



9.61364 
9.61400 
9.61436 
9.61472 
9.61508 



41251 
285 
3 J 9 
353 
387 



9.61544 
9.61579 
9.61615 
9.61651 
9.61687 



41421 
455 
49o 
524 
558 



9.61722 
9.61758 
9.61794 
9.61830 
9.61865 



41592 
626 
660 
694 
728 



9.61901 
9.61936 
9.61972 
9.62008 
9.62043 



41763 
797 
831 
865 

899 



9.62079 
9.62114 
9.62150 
9.62185 
9.62221 



41933 
968 

42002 
036 
070 



9.62256 
9.62292 
9.62327 
9.62362 
9.62398 



42105 
139 
173 
207 
242 



9-62433 
9.62468 
9.62504 

9-62539 
9.62574 



42276 
310 
345 
379 
413 
447 



9.62609 
9.62645 
9.62680 
9.62715 
9.62750 
9.62785 



0-39359 
0.39323 
0.39286 
0.39250 
0-39214 



2.4751 
73o 
709 
689 
668 



0.39177 
O.39141 
0.39105 
O.39069 
0-39033 



2.4648 
627 
606 
586 
566 



0.38996 
O.38960 
0.38924 
O.38888 
0-38852 



2-4545 
525 
504 
484 

464 



0.38816 
0.38780 
0.38744 
0.38708 
0.38672 



2.4443 
423 
403 
383 
362 



0.38636 
0.38600 
0.38564 
0.38528 
0.38492 



2.4342 
322 
302 
282 
262 



0.38456 
0.38421 
0.38385 
0.38349 
0.38313 



2.4242 
222 
202 
182 
162 



0.38278 
0.38242 
0.38206 
0.38170 
0.38135 



2.4142 
122 
102 
083 
063 



0.38099 
0.38064 
0.38028 
0.37992 
0-37957 



2.4043 
023 
004 

2.3984 
964 



0.37921 
0.37886 
0.37850 
0-378I5 
0-37779 



2-3945 
925 
906 
886 
867 



0-37744 
0.37708 

0-37673 
O.37638 
0.37602 



2.3847 
828 
808 
789 
770 



0.37567 
0.37532 
0.37496 
0.37461 
0.37426 



2.375o 
73i 
712 

693 
673 



0-37391 
0-37355 
0.37320 
0.37285 
0.37250 
0.37215 



2.3654 
635 
616 

597 

578 
559 



Nat. COS Log. d. Nat. Sin Log. d. Nat. CotLog. c.d. Log. Tan Nat 

w° 



23° 



f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat Tan Log.jcd. Log. Cot Nat. 







39073 9-59188 




92050 9.96403 


5 


42447 9-62785 QC 
482 9.62820 6b 
516 9.62855 85 
551 9.62890 35 
585 9.62926 f _ 

42619 9.62961 35 
654 9.62996 f 
688 9.63031 35 
722 9.63066 ;~ 


0.37215 2.3559 


60 


I 


100 9.59218 


3° 


o39 9-96397 


5 

5 
6 


0.37180 539 


59 


2 


127 9.59247 


29 
30 
30 
29 


028 9.96392 


0-37145 520 


58 


3 


153 9-59277 


016 9.96387 


0.37110 501 


57 


4 


180 9.59307 


005 9.96381 


5 
6 


0.37074 483 


5o 
55 


5 


39207 9.59336 


91994 9.96376 


0-37039 2.3464 


6 


234 9-59366 


3° 


982 9.96370 


5 

5 
5 


0.37004 445 


54 


7 


260 9.59396 


3 U 


971 9.96365 


0.36969 426 


53 


8 


287 9.59425 


29 


959 9-96360 


0.36934 407 


52 


9 


314 9-59455 


3° 
29 


948 9-96354 


5 
6 


757 9-63101 


34 


0.36899 388 


5i 
50 


10 


39341 9.59484 


91936 9.96349 


42791 9.63135 


0.36865 2.3369 


ii 


3 6 7 9595 J 4 


3° 


925 9.96343 


5 

5 
6 


826 9.63170 ;» 


0.36830 351 


49 


12 


394 9-59543 


29 


914 9-96338 


860 9.63205 35 


0-36795 332 


48 


13 


421 9-59573 


3° 


902 9.96333 


894 9.63240 ii 


0.36760 313 


47 


14 


448 9.59602 


29 

30 


891 9.96327 


5 
5 


929 9.63275 


■50 

35 


0.36725 294 


46 
45 


15 


39474 9-59632 


91879 9.96322 


42963 9-633 J o 


0.36690 2.3276 


16 


501 9.59661 


29 


868 9.96316 


5 
6 


998 9.63345 f» 


0.36655 257 


44 


17 


528 9.59690 


29 


856 9-96311 


43032 9.63379 ** 


0.36621 238 


43 


18 


555 9-59720 


3° 


845 9-96305 


5 
6 

5 

5 
g 


067 9.63414 •» 


0.36586 220 


42 


19 


581 9.59749 


29 

29 


833 9-96300 


101 9.63449 


3D 

35 


0.36551 201 


4i 
40 


20 


39608 9-59778 


91822 9.96294 


43I3 6 9-63484 


0.36516 2.3183 


21 


°35 9-598o8 


3° 


810 9.96289 


170 9-63519 £ 


0.36481 164 


39 


22 


661 9.59837 


29 


799 9.96284 


205 9-63553 it 


0.36447 146 


38 


23 


688 9.59866 


29 


787 9.96278 


5 
6 

5 


239 9-63588 | ii 


0.36412 127 


37 


24 


715 9-59895 


29 
29 


775 9.96273 


274 9-63623 


33 

lit 


0-36377 io 9 
0.36343 2.3090 


30 
35 


25 


39741 9.59924 


91764 9.96267 


43308 9-63657 


26 


768 9.59954 


3° 


752 9.96262 


. 343 9-63692 : f. 


0.36308 072 


34 


27 


795 9-59983 


29 


741 9.96256 


5 
5 


378 9-63726 JJ 


0.36274 053 


33 


28 


822 9.60012 


29 


729 9.96251 


412 9.63761 ii 


0.36239 035 


32 


29 


848 9.60041 


29 
29 


718 9.96245 


5 
5 


447 9-63796 


34 


0.36204 017 
0.36170 2.2998 


3i 
30 


30 


39875 9.60070 


91706 9.96240 


43481 9.63830 


31 


902 9.60099 


29 


694 9.96234 


5 
6 

5 
6 

5 


516 9.63865 i £> 


0-36I35 98o 


29 


32 


928 9.60128 


29 


683 9.96229 


550 9.63899 34 


0.36101 962 


28 


33 


955 9-6oi57 


29 


671 9.96223 


585 9-63934 ii 


0.36066 944 


27 


34 


982 9.60186 


29 
29 


660 9.96218 


620 9.63968 


3^ 

35 


0.36032 925 


26 
25 


35 


40008 9.60215 


91648 9.96212 


43654 9.64003 


0-35997 2.2907 


36 


035 9.60244 


29 


636 9.96207 


689 9.64037 


34 
35 


0-35963 88 9 


24 


37 


062 9.60273 


29 


625 9.96201 




724 9.64072 


0.35928 871 


23 


38 


088 9.60302 


29 


613 9.96196 


5 


758 9.64106 


34 


0.35894 853 


22 


39 
40 


115 9.60331 


29 
28 


601 9.96190 


5 
5 


793 964140 


34 
35 


0.35860 835 


21 


40141 9.60359 


91590 9.96185 


43828 9.64175 


0.35825 2.2817 


20 


41 


168 9.60388 


29 


578 9.96179 


5 
6 


862 9.64209 


34 


0.35791 799 


IQ 


42 


195 9.60417 


29 


566 9.96174 


897 9.64243 


34 


0-35757 78i 


18 


43 


221 9.60446 


29 
28 
29 


555 9-96i68 


5 


932 9.64278 


35 


0.35722 763 


17 


44 


248 9.60474 


543 9.96162 


5 
5 


966 9.64312 


34 
34 


0.35688 745 


16 


45 


40275 9.60503 


91531 9-96I57 


44001 9.64346 


0.35654 2.2727 


15 


46 


301 9.60532 


29 


519 9.96151 




036 9.64381 


35 


0-35619 7°9 


14 


47 


328 9.60561 


29 

28 


508 9.96146 


5 


071 9.64415 


34 


0.35585 691 


13 


48 


355 9.60589 




496 9.96140 




105 9.64449 


34 


0.35551 673 


12 


49 


381 9.60618 


29 
28 


484 996135 


5 
6 
6 


140 9.64483 


34 
34 


0-355I7 655 


11 


50 


40408 9.60646 


91472 9.96129 


44175 9.64517 


0.35483 2.2637 


10 


Si 


434 9.60675 


29 


461 9.96123 




210 9.64552 a 


0.35448 620 


9 


52 


461 9.60704 


29 
28 


449 9.96118 


5 


244 9-64586 


3-4- 

34 


0.35414 602 


8 


S3 


488 9.60732 




437 9-96ii2 




279 9.64620 


o.3538o 584 


7 


54 


514 9.60761 


29 
28 
29 
28 


425 9.96107 


5 
6 
6 


314 9.64654 


34 
34 
34 


0-35346 566 


6 


55 


40541 9.60789 


91414 9.96101 


44349 9-64688 


0.35312 2.2549 


5 


56 


567 9.60818 


402 9.96095 


5 
5 


384 9.64722 


0.35278 53i 


4 


57 


594 9.60846 




390 9.96090 


418 9.64756 


34 


0.35244 5i3 


3 


5« 


621 9.60875 


29 


378 9.96084 




453 9-64790 


34 


0.35210 496 


2 


59 


647 9.60903 


28 


366 9.96079 


5 
6 


488 9.64824 


34 
34 


0.35176 478 


1 


60 


674 9.60931 




355 9-96073 




523 9.64858 


0.35142 460 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. c.d.|Log.Tan Nat. 


/ 



66 



24 c 



Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog. c.d. Log. Cot Nat 



40674 
700 
727 

753 
780 



9.60931 
9.60960 
9.60988 
9.61016 
9.61045 



40806 

833 
860 
886 
913 



9.61073 
9.61101 
9.61 129 
9.61 158 
9.61186 



40939 
966 
992 

41019 
04S 



9.61214 
9.61242 
9.61270 
9.61298 
9.61326 



41072 
098 
125 
151 
178 



9-01354 
9.61382 
9.61411 
9.61438 
9.61466 



41204 
231 

257 
284 
310 



9.61494 
9.61522 
9.61550 
9.61578 
9.61606 



41337 
363 
390 
416 
443 



9.61634 
9.61662 
9.61689 
9.61717 
9.61745 



41469 
496 
522 
549 
575 



9.61773 
9.61800 
9.61828 
9.61856 
9.61883 



41602 
628 

655 
681 
707 



9.61911 
9.61939 
9.61966 
9.61994 
9.62021 



41734 
760 

787 

8i3 
840 



9.62049 
9.62076 
9.62104 
9.62131 
9.62159 



41866 
892 
919 

945 
972 



9.62186 
9.62214 
9.62241 
9.62268 
9.62296 



42024 

051 
077 
104 



9.62323 
9.62350 
9.62377 
9.62405 
9.62432 



42130 
156 
183 
209 

235 
262 



9.62459 
9.62486 
9.62513 
9.62541 
9.62568 
9-62595 



91355 
343 
33i 
319 
3°7 



9.96073 
9.96067 
9.96062 
9.96056 
9.96050 



91295 
283 
272 
260 
248 



9.96045 
9.96039 
9.96034 
9.96028 
9.96022 



91236 
224 
212 
200 
188 



9.96017 
9.9601 1 
9.96005 
9.96000 
9-95994 



91176 
164 
152 
140 
128 



9.95988 
9.95982 
995977 
9-95971 
9-95965 



91116 
104 
092 
080 



9.95960 

9-95954 
9.95948 
9.95942 
9-95937 



91056 
044 
032 
020 
008 



9-95931 
9-95925 
9.95920 

9-959I4 
9.95908 



90996 
984 
972 
960 
948 



9.95902 

9-95897 
9.95891 

9.95885 
995879 



90936 
924 
911 



9-95873 
9.95868 
9.95862 
9-95856 
9-95850 



90875 
863 
851 
839 
826 



9.95844 
9-95839 
9-95833 
9.95827 
9.95821 



90814 
802 
790 
778 
766 



9-958I5 
9.95810 
9.95804 
9-95798 
9-95792 



90753 
741 
729 
717 
704 



9.95786 
9.95780 
9-95775 
995769 
995763 



680 
668 
655 
643 
631 



9-95757 
9-95751 
9-95745 
9-95739 
9-95733 
9.95728 



44523 
558 
593 
627 
662 



9.64858 
9.64892 
9.64926 
9.64960 
9.64994 



44697 
732 
767 
802 
837 



9.65028 
9.65062 
9.65096 
9.65130 
9.65164 



44872 
907 
942 
977 

45012 



9-65I97 
9.65231 
9.65265 
9.65299 
9-65333 



45047 
082 
117 
152 
187 



9.65366 
9.65400 

9-65434 
9.65467 

9-6550I 



45222 

257 
292 

327 
362 



9-65535 
9.65568 
9.65602 
9.65636 
9.65669 



45397 
432 
467 
5°2 
538 



9-65703 
9-65736 
9-65770 
9.65803 
9-65837 



45573 
608 

643 
678 

713 



9.65870 
9.65904 

9-65937 
9.65971 
9.65004 



45748 
784 
819 

854 



9.66038 
.9.66071 
9.66104 
9.66138 
9.66171 



45924 
960 

995 

46030 

065 



9.66204 
9.66238 
9.66271 
9.66304 
9-66337 



46101 
136 
171 
206 
242 



9.66371 
9.66404 
9.66437 
9.66470 
9.66503 



46277 
312 
348 
383 
418 



9-66537 
9.66570 
9.66603 
9.66636 
9.66669 



40454 
489 
525 
560 

595 
631 



9.66702 

9-66735 
9.66768 
9.66801 
9.66834 
9.66867 



0.35142 
0.35108 

0.35074 
0.35040 
0.35006 



2.2460 

443 
425 
408 

390 



0.34972 
0.34938 
0.34904 
O.34870 
0.34836 



2.2373 
355 
338 

320 

303 



0.34803 
0.34769 
0-34735 
0.34701 
0.34667 



2.2286 
268 
251 
234 
216 



0.34634 
0.34600 
0.34566 
0-34533 
Q-34499 



2.2199 
182 
165 
148 
130 



0.34465 
0.34432 
0.34398 
0.34364 
034331 



2.2113 
096 
079 
062 
045 



0.34297 
0.34264 
0.34230 
0.34197 
0.34163 



2.2028 

on 

2.1994 

977 
960 



0.34130 
O.34096 
0.34063 
0.34029 
0.33996 



2.1943 
926 
909 
892 
876 



0.33962 
033929 
O.33896 
0.33862 
0.33829 



2.1859 
842 
825 
808 
792 



0.33796 
0.33762 
0.33729 
0.33696 
0.33663 



2.1775 
758 
742 

725 
708 



0.33629 
0.33596 
0.33563 
0-33530 
0-33497 



2.1692 
675 
659 
642 
625 



0.33463 
0.33430 
0-33397 
0.33364 
0.33331 



2.1609 

592 
570 
560 

543 



0.33298 
0.33265 
0.33232 
0.33199 
0.33166 

0.33133 



2.1527 
5io 
494 
478 
461 
445 



Nat.CoSLog. d. 



Nat. Sin Log. d. 

05 



Nat. Cot Log. c.d. Log.TanNat 



25° 



f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







42262 9.62595 


27 
27 
27 
27 
27 


90631 9.95728 


6 


46631 9.66867 


33 
33 
33 
33 
33 


0-33I33 2.1445 


60 


I 


288 9.62622 


618 9.95722 


5 


666 9.66900 


0.33100 429 


59 


2 


315 9.62649 


606 9.95716 


6 


702 9.66933 


0.33067 4i3 


58 


3 


341 9.62676 


594 9-95710 


6 


737 9-66966 


0.33034 396 


57 


4 
5 


367 9.62703 


582 9.95704 


6 

6 


772 9.66999 


0.33001 380 


56 


42394 9-62730 


90569 9.95698 


46808 9.67032 


0.32968 2.1364 


55 


6 


420 9.62757 


27 

27 
27 
27 
27 


557 9-95692 


6 


843 9.67065 


33 
33 
33 

32 
33 
33 
33 
33 


0.32935 348 


54 


7 


446 9.62784 


545 9-95686 


6 


879 9.67098 


0.32902 332 


53 


8 


473 9.6281 1 


532 9.95680 


6 


914 9.67131 


0.32869 315 


52 


9 


499 9.62838 


520 9.95674 


6 
5 


950 9.67163 


0.32837 299 


5i 
50 


10 


42525 9.62865 


90507 9.95668 


46985 9.67196 


0.32804 2.1283 


ii 


552 9.62892 


°6 


495 9-95663 


47021 9.67229 


0.32771 267 


49 


12 


578 9.62918 


27 


483 995657 


6 


056 9.67262 


0.32738 251 


48 


13 


604 9.62945 


470 9.95651 


6 


092 9.67295 


0.32705 235 


47 


14 


631 9.62972 


27 


458 9-95645 


6 
6 


128 9.67327 


32 
33 


0.32673 219 


46 
45 


15 


42657 9.62999 


90446 9.95639 


47163 9.67360 


0.32640 2.1203 


16 


683 9.63026 


27 

05 


433 9-95633 


6 


199 9-67393 


33 

33 
32 
33 
33 
32 


0.32607 187 


44 


i? 


709 9.63052 


27 
27 
27 
26 


421 9.95627 


6 


234 9.67426 


0.32574 171 


43 


18 


736 9.63079 


408 9.95621 


fi 


270 9.67458 


0.32542 155 


42 


19 


762 9.63106 


396 9.95615 


6 
6 


305 9-6749I 


0.32509 139 


4i 
40 


20 


42788 9.63133 


90383 9-95609 


47341 9.67524 


0.32476 2.1 123 


21 


815 9.63159 


27 
27 
°6 


371 9-95603 


f, 


377 9-67556 


0.32444 107 


39 


22 


841 9.63186 


358 9-95597 


6 


412 9.67589 


^ 0.3241 1 092 
o 2 0.32378 076 


38 


23 


867 9.63213 


346 9-95591 


6 


448 9.67622 


37 


24 


894 9.63239 


27 
06 


334 9-95585 


6 
6 


483 9.67654 


3 Z 


0.32346 060 


30 
35 


25 


42920 9.63266 


90321 9-95579 


47519 9.67687 


33 


0.32313 2.1044 


26 


946 9.63292 


27 
06 


309 9-95573 


f, 


555 9-677I9 00 1 0.32281 028 
590 9.67752 g 0.32248 013 
626 9.67785 £J ! 0.32215 2.0997 


34 


27 


972 9- 6 33 I 9 


296 9-95567 


6 


33 


28 


999 9-63345 


27 
26 
27 
^6 


284 9-95561 


(\ 


32 


29 


43025 9.63372 


271 9-95555 


6 
6 


662 9.67817 


33 


0.32183 981 


3i 


30 


43051 9-63398 


90259 9-95549 


47698 9.67850 


0.32150 2.0965 


30 


31 


077 9.63425 


246 9-95543 


fi 


733 9-67882 j£ | 0.32118 950 
769 9-679I5 02 0-32085 934 
805 9.67947 1 ^ 0.32053 918 


29 


32 


104 9-6345I 


27 

^6 


233 9-95537 


6 


28 


33 


130 9.63478 


221 9-95531 


fi 


27 


34 


156 9-63504 


27 
06 


208 9.95525 


6 
6 


840 9.67980 


32 


0.32020 903 


26 
25 


35 


43182 9.63531 


90196 9.95519 


47876 9.68012 


0.31988 2.0887 


36 


209 9-63557 


°6 


183 9-955I3 


6 


912 9.68044 ' ^ 0.31956 872 
948 9.68077 ^ 0.31923 856 


24 


37 


235 9-63583 


27 
06 


171 9-95507 


7 
6 


23 


3« 


261 9,63610 


158 9-955oo 


984 9.68109 ^ 0.31891 840 


22 


39 


287 9.63636 


26 

27 
^6 


146 9.95494 
90133 9.95488 


6 
6 


48019 9.68142 


03 
32 


0.31858 825 


21 


40 


43313 9-63662 


48055 9.68174 


0.31826 2.0809 


20 


4i 


340 9.63689 


120 9.95482 


f, 


091 9.68206 ;£ | 0.31794 794 
127 9.68239 g 0.31761 778 
163 9.68271 1 ^ o 0.31729 763 


IQ 


42 


366 9-637I5 


06 


108 9.95476 


6 


18 


43 


392 9.63741 


°6 


095 9-95470 


6 


17 


44 


418 9.63767 


27 
"6 


082 9.95464 


6 
6 


198 9.68303 


33 


0.31697 748 


16 


45 


43445 9-63794 


90070 9.95458 


48234 9.68336 


0.31664 2.0732 


15 


46 


471 9.63820 


°6 


o57 9-95452 


6 


270 9.68368 D Z 0.31632 717 
306 9.68400 6 ! 0.31600 701 


14 


47 


497 9-63846 


26 


045 9-95446 


6 


13 


48 


523 9.63872 


"6 


032 9.95440 


6 


342 9.68432 : 3 o j 0.31568 686 


12 


49 


549 9-63898 


26 


019 9-95434 


7 


378 9.68465 


32 


0.3I535 671 


11 


50 


43575 9-63924 


90007 9.95427 


48414 9.68497 


0.31503 2.0655 


10 


5i 


602 9.63950 


"6 


89994 9-95421 


f, 


450 9-68529 ^ 
486 9.68561 | f 
521 9.68593 j ^ 


0.31471 640 


9 


52 


628 9.63976 


~6 


981 9-954I5 


fi 


0.31439 625 


8 


53 


654 9.64002 


^6 


968 9.95409 


6 


0.31407 609 


7 


54 


680 9.64028 


26 
°6 


956 9-95403 


6 


557 9.68626 


00 
32 


0.31374 594 


6 


55 


43706 9.64054 


89943 9-95397 


48593 9.68658 | 


0.31342 2.0579 


5 


56 


733 9.64080 


°6 


93o 9-95391 


7 
6 


629 9.68690 a* 0.31310 564 
665 9.68722 * 0.31278 549 
701 9.68754 | 0.31246 533 
737 9-68786 \ 0.31214 518 


4 


57 


759 9.64106 


°6 


918 9.95384 


3 


5* 


785 9.64132 


06 


905 9-95378 


6 


2 


S9 


811 9.64158 


26 


892 9-95372 


5 


1 


60 


837 9.64184 




. 879 9-95366 




773 9.68818 6 0.31 182 503 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. c.d. Log. Tan Nat. 


/ 



64 ( 









26 


D 








t 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







43837 9-64184 


26 


89879 9-95366 


5 


48773 9.68818 




0.31 182 2.0503 


60 


I 


863 9.64210 


26 


867 9-9536o 


6 

6 


809 9.68850 


S 2 


0.31150 488 


59 


2 


809 9.64236 


26 


854 9-95354 


845 9.68882 


3 2 

32 
32 
32 
32 
32 
32 


0.31 1 18 473 


58 


3 


916 9.64262 


26 


841 9-95348 




881 9.68914 


0.31086 458 


57 


4 


942 9.64288 


25 
26 


828 9.95341 


7 
6 
6 


917 9.68946 


0.31054 443 


56 
55 


5 


43968 9.64313 


89816 9.95335 


48953 9.68978 


0.31022 2.0428 


6 


994 9-64339 


"6 


803 9-95329 


5 


989 9.69010 


0.30990 413 


54 


7 


44020 9.64365 


26 


790 9.95323 


6 


49026 9.69042 


0.30958 398 


53 


8 


046 9.64391 


26 


777 9-953 J 7 




062 9.69074 


0.30926 383 


5 2 


9 
10 


072 9.64417 


25 
26 


764 9-95310 


7 
6 
5 


098 9.69106 


32 
32 
32 


0.30894 368 


5i 


44098 9.64442 


89752 9-95304 


49134 9.69138 


0.30862 2.0353 


50 


ii 


124 9.64468 


26 


739 9-95298 


5 


170 9.69170 


0.30830 338 


49 


12 


151 9-64494 


25 

26 


726 9.95292 


6 


206 9.69202 


32 


0.30798 3 2 3 


48 


13 


177 9.64519 


713 9.95286 


242 9.69234 


32 

32 
32 
31 
32 


0.30766 308 


47 


14 
15 


203 9.64545 


26 

25 
"6 


700 9.95279 


7 
6 
5 


278 9.69266 


0.30734 2 93 


4 b 
45 


44229 9.64571 


89687 9.95273 


49315 9-69298 


0.30702 2.0278 


16 


2 55 9-64596 


674 9.95267 


6 


351 9.69329 


O.30671 263 


44 


17 


281 9.64622 


25 
26 


662 9.95261 




387 9.69361 


0.30639 248 


43 


18 


307 9.64647 


649 9-95254 


7 
5 


423 9-69393 


32 

32 
32 
31 


0.30607 233 


42 


19 


333 9-64673 


25 

~6 


636 9.95248 


6 
6 


459 9-69425 


0-30575 219 


4i 
40 


20 


44359 9-64698 


89623 9.95242 


49495 9-69457 


0.30543 2.0204 


21 


385 9.64724 


25 
26 


610 9.95236 




532 9.69488 


0.30512 189 


39 


22 


411 9.64749 


597 9-95229 


7 
5 


568 9.69520 


3 2 
32 


0.30480 174 


38 


23 


437 9-64775 


25 
26 

25 
26 


584 9.95223 


6 
6 


604 9.69552 


O.30448 160 


37 


24 


464 9.64800 


571 9-952I7 


640 9.69584 


32 

3i 
32 


0.30416 145 


36 
35 


25 


44490 9.64826 


89558 9-952H 


49677 9-69615 


0.30385 2.0130 


26 


516 9.64851 


545 9-95204 


7 
5 


713 9.69647 


0.30353 "5 


34 


27 


542 9.64877 


25 

25 
26 

25 
25 
26 


532 9-95I98 


5 


749 9.69679 


S 2 


0.30321 101 


33 


28 


568 9.64902 


519 9-95I92 




786 9.69710 


3 1 


0.30290 086 


32 


29 


594 9-64927 


506 9.95185 


7 
6 
6 


822 9.69742 


32 
32 


0.30258 072 
0.30226 2.0057 


3i 
30 


30 


44620 9.64953 


89493 9-95*79 


49858 9.69774 


3i 


646 9.64978 


480 9-95 J 73 


6 


894 9.69805 


3 1 


0.30195 042 


29 


32 


672 9.65003 


467 9.95167 


7 
6 


931 9.69837 


3 2 
3i 


0.30163 028 


28 


33 


698 9.65029 


25 

25 

25 
26 


454 9-95 J 6o 


967 9.69868 


0.30132 013 


27 


34 
35 


724 9.65054 


441 9.95154 


6 


50004 9.69900 


3 2 

32 
31 
32 
3i 


0.30100 1.9999 


2b 
25" 


4475° 9-65079 


89428 9.95148 


50040 9.69932 


0.30068 1.9984 


36 


776 9.65104 


415 9.95141 


7 
6 


076 9.69963 


0.30037 970 


24 


37 


802 9.65130 


25 


402 9.95135 


6 


113 9.69995 


0.30005 955 


23 


3* 


828 9.65155 


389 9-95129 




149 9.70026 


0.29974 941 


22 


39 
40 


854 9.65180 


25 
25 
25 
25 
26 


376 9.95122 


7 
6 
5 


185 9.70058 


3 2 
3i 


0.29942 926 


21 
20 


44880 9.65205 


89363 9-95 IJ 6 


50222 9.70089 


0.29911 1.9912 


4i 


906 9.65230 


350 9.95110 




258 9.70121 


32 


0.29879 897 


19 


42 


932 9.65255 


337 9-95!03 


7 
5 


295 9-70I52 


3 1 


0.29848 883 


18 


43 


958 9.65281 




324 9-95097 




331 9.70184 


32 

3i 
32 
3i 


0.29816 868 


17 


44 


984 9.65306 


2 5 

25 
25 


311 9.95090 


7 
6 
6 


368 9.70215 


0.29785 854 


lb 
15" 


45 


45010 9-6533I 


89298 9.95084 


50404 9.70247 


0.29753 1-9840 


4 b 


036 9-65356 


285 9-95078 




441 9.70278 


0.29722 825 


14 


47 


062 9.65381 


25 


272 9-9507 1 


7 
5 


477 9-70309 


3 1 


0.29691 811 


13 


48 


088 9.65406 


25 
25 

25 


259 9-95065 


5 


514 9.70341 


32 


O.29659 797 


12 


49 
50 


114 9.65431 


245 9-95059 


7 
5 


55° 9-70372 


3 1 
32 
3i 
3i 


0.29628 782 


11 


45140 9.65456 


89232 9.95052 


50587 9.70404 


0.29596 1.9768 


10 


Si 


166 9.65481 


2 5 
25 
25 
25 
24 


219 9.95046 




623 9-70435 


0.29565 754 


9 


52 


192 9.65506 


206 9.95039 


7 
5 


660 9.70466 


0.29534 740 


8 


53 


218 9.65531 


193 9-95033 


6 


696 9.70498 


32 
3i 
3i 


0.29502 725 


7 


54 


243 9-65556 


180 9.95027 


7 
6 


733 9-70529 


0.29471 711 


6 


55 


45269 9.65580 


89167 9.95020 


50769 9.70560 


0.29440 1.9697 


5 





295 9-65605 


25 


*53 9-95014 




806 9.70592 


32 

3i 


0.29408 683 


4 


57 


321 9.65630 


25 


140 9.95007 


7 
5 


843 9.70623 


0.29377 669 


3 


# 


347 9-65655 


25 
25 


127 9.95001 


6 


879 9-70654 


3 1 
3i 


0.29346 654 


2 


IS 


373 9-65680 


114 9.94995 




916 9.70685 


0.29315 640 


1 


399 9-65705 


25 


101 9.94988 | ' 


953 9-707I7 


3 2 


0.29283 626 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


r 










6? 


O 















27° 








f 


Nat. Sin Log. d. 


Nat. COS Log. d. |Nat Tan Log. 


c.d. 


Log. Cot Nat. 







45399 O-65705 


24 

25 
25 
25 
24 

25 
25 


89101 9.94988 


6 


50953 9-707I7 


31 
31 
31 
31 
32 
31 


0.29283 1.9626 


60 


I 


425 9-65729 


087 9.94982 




989 9.70748 


0.29252 612 


59 


2 


45i 9-65754 


074 9-94975 


7 
6 


51026 9.70779 


0.29221 598 


58 


3 


477 9-65779 


061 9.94969 




063 9.70810 


0.29190 584 


57 


4 


503 9.65804 


048 9.94962 


7 
6 

7 
6 


°99 9.70841 


0.29159 570 


56 


5 


45529 9-65828 


89035 9-94956 


5 1 136 9.70873 


0.29127 1.9556 


55 


b 


554 9-65853 


021 9.94949 


173 9.70904 


0.29096 542 


54 


7 


580 9.65878 


008 9.94943 




209 9.70935 £ 
246 9.70966 | ** 


0.29065 528 


53 


8 


60b 9.65902 


24 
25 
25 
24 
25 


88995 9-94936 


7 

5 


0.29034 514 


52 


9 


632 9.65927 


981 9.94930 


7 
6 


283 9-70997 


D-i 


0.29003 500 


5i 
50 


10 


45658 9-65952 


88968 9.94923 


51319 9.71028 


3 1 
31 
31 
3i 


0.28972 1.9486 


ii 


b84 9.65976 


955 9-949I7 


6 


356 9.71059 


0.28941 472 


49 


12 


710 9.66001 


942 9.9491 1 




393 9-7I090 


0.28910 458 


48 


13 


736 9.66025 


24 


928 9.94904 


7 
6 

7 
5 


430 9.71121 


0.28879 444 


47 


14 


762 9.66050 


25 
25 


915 9.94898 


467 9.71 153 


32 
3i 


0.28847 430 


46 


15 


45787 9.66075 


88902 9.94891 


51503 9.71 184 


0.28816 1.9416 


45 


ib 


813 9.66099 


24 
25 


888 9.94885 




540 9.71215 £ 0.28785 402 
577 9-71246 3 0.28754 388 


44 


17 


839 9.66124 


875 9.94878 


7 


43 


18 


865 9.66148 


24 
25 
24 
24 


862 9.94871 


7 
6 

7 
5 


614 9.71277 £ | 0.28723 375 


42 


19 


891 9.66173 


848 9.94865 


651 9-7 x 308 


J- 1 - 

31 


0.28692 361 


4i 


20 


45917 9.66197 


88835 9-94858 


51688 9.71339 


0.28661 1.9347 


40 


21 


942 9.66221 


822 9.94852 




724 9-7I370 2: 


0.28630 333 


39 


22 


9b8 9.66246 


25 


808 9.94845 


7 
b 


761 9.71401 


J- 1 - 

30 


0.28599 319 


38 


23 


994 9.66270 


24 


795 9-94839 


798 9-7I43I 


0.28569 306 


37 


24 


46020 9.66295 


2 5 
24 


782 9.94832 


7 

b 


835 9.71462 


3 1 


0.28538 292 


30 


25 


4604b 9.66319 


88768 9.94826 


51872 9.71493 


3 1 


0.28507 1.9278 


35 


2b 


072 9.66343 


24 
25 


755 9-948I9 


7 

5 


909 9-7*524 


3 1 1 0.28476 265 
£ 0.28445 251 


34 


27 


097 9.66368 


741 9.94813 




946 9-7I555 


33 


28 


123 9.66392 


24 


728 9.94806 


7 


9 8 3 9-7 I 586 ** j 0.28414 237 


32 


29 


149 9.66416 


24 
25 


715 9-94799 


7 
b 


52020 9.71617 


3i 


0.28383 223 


3i 


30 


46175 9.66441 


88701 9.94793 


52057 9-7*648 


0.28352 1. 9210 


30 


31 


201 9.66465 


24 


688 9.94786 


7 
6 


094 9.71679 


3 1 


0.28321 196 


29 


32 


226 9.66489 


24 


6 74 9-94780 


131 9-7I709 £ 


0.28291 183 


28 


33 


252 9.66513 


24 


661 9.94773 


7 

b 

7 


168 9.71740 2* 


0.28260 169 


27 


34 


278 9.66537 


24 
25 
24 


647 9-94767 


205 9.71771 


j 1 
3i 


0.28229 155 


2b 


35 


46304 9.66562 


88b34 9.94760 


52242 9.71802 


0.28198 1.9142 


25 


36 


330 9.66586 


620 9.94753 


7 
6 


279 9-7I833 


3 1 


0.28167 128 


24 


37 


355 9.66610 


24 


607 9-94747 


316 9.71863 


3° 


0.28137 115 


23 


38 


381 9.66634 


24 


593 9-94740 


7 
6 

7 


353 9-7I894 


3 1 


0.28106 101 


22 


39 
40 


407 9.66658 


24 
24 


580 9.94734 


390 9.71925 ! ^ 
52427 9-7*955 lj 


0.28075 088 


21 


46433 9.66682 


885bb 9-94727 


0.28045 1.9074 


20 


41 


458 9.66706 


24 
25 


553 9-94720 


7 
6 


464 9.7I986 ^ ! 0.28014 06l 


19 


42 


484 9.66731 


539 9-947I4 


50I 9.72017 


* O.27983 047 


18 


43 


510 9-66755 


24 


526 9-94707 


7 


538 9.72O48 


% 0.27952 034 


17 


44 


536 9.66779 


24 
24 


512 9.94700 


7 

6 


575 9-72078 


3° 


0.27922 020 


ib 


45 


465b! 9.66803 


88499 9.94694 


52bi3 9.72109 


3 1 


0.27891 1.9007 


15 


4 b 


587 9.66827 


24 


485 9.94687 


7 


650 9.72140 


3 1 


0.27860 1.8993 


14 


47 


613 9.66851 


24 
24 


472 9.94680 


7 
5 


687 9.72170 


3° 


0.27830 980 


13 


48 


639 9.66875 


458 9-94674 




724 9.72201 | ; 


0.27799 967 


12 


49 


664 9.66899 


24 
23 
24 

24 


445 9-94667 


7 

7 
5 


761 9.72231 


3 W 
3i 
3i 
30 
3i 


0.27769 953 


11 


50 


46690 9.66922 


88431 9.94660 


52798 9.72262 


0.27738 1.8940 


10 


5i 


71b 9.66946 


417 994654 




836 9.72293 


0.27707 927 


9 


52 


742 9.66970 


404 9.94647 


7 


873 9-72323 


0.27677 913 


8 


<^3 


7b7 9.66994 


24 


390 9.94640 


7 
6 

7 


910 9-72354 


0.27646 900 


7 


54 


793 9-67018 


24 
24 


377 9-94634 


947 9-72384 


3° 


0.27616 887 


6 


55 


46819 9.67042 


883b3 9.94627 


52985 9.72415 


3 1 


, 0.27585 1.8873 


5 


.0 


844 9.67066 


24 


349 9-94620 


7 
5 


53022 9.72445 £ | 0.27555 860 

059 9.72476 i ^ 1 0.27524 847 
096 9.72506 ^ 0.27494 834 
134 9.72537 1 * 0.27463 820 


4 


57 


870 9.67090 


24 
23 


336 9.94614 




3 


# 


896 9.67113 


322 9.94607 


7 


2 


^ 9 o 


921 9-67I37 


JJi 308 9-94600 


7 


1 


947 9.67161 


^1 295 9.94593 


7 171 9-72567 1 D " 1 0.27433 807 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log 


cd.JLog.Tan Nat. 


/ 



62 ( 



Nat. Sin Log. d. 



46947 
973 
999 

47024 
050 



9.67161 
9.67185 
9.67208 
9.67232 
9.67256 



47076 
101 
127 

153 

178 



9.67280 

9-67303 
9.67327 

9-67350 
9-67374 



47204 
229 

255 
281 
306 



9.67398 
9.67421 

9-67445 
9.67466 
9.67492 



47332 
358 
383 
409 

434 



9-67515 
9-67539 
9.67562 
9.67586 
9.67609 



47460 
486 
5ii 
537 
562 

47588 
614 
639 
665 
690 



9.67633 
9.67656 
9.67680 
9.67703 
9.67726 

9-67750 
9-67773 
9.67796 
9.67820 
9.67843 



47716 
741 
767 

793 
818 



9.67866 
9.67890 
9.67913 
9.67936 
9-67959 



H4 



920 
946 



9.67982 
9.68006 
9.68029 
9.68052 
9.68075 



47971 

997 

48022 

048 

073 



9.68098 
9.68121 
9.68144 
9.68167 
9.68190 



48099 
124 
ISO 
175 
201 



9.68213 
9.68237 
9.68260 
9.68283 
9.68305 



48226 
252 
277 
303 
328 

48354 
379 
405 
430 
456 
481 



9.68328 
9.68351 
9-68374 
9.68397 
9.68420 
9.68443 
9.68466 
9.68489 
9.68512 
9-68534 
9-68557 



2_8 

Nat. COS Log. d. 



88295 
281 
267 

254 
240 



9-94593 
9-94587 
9.94580 

9-94573 
9-94567 



88226 

2T 3 
I99 

185 
I72 



9.94560 

9-94553 
9.94546 
9.94540 
9-94533 



S158 
144 
130 
117 
103 



9.94526 
9-945 J 9 
9-945 I 3 
9.94506 
9.94499 



075 

062 

048 

_?34 

88020 

006 

87993 

979 

965 



9.94492 
9.94485 
9.94479 
9.94472 
9.94465 



9.94458 
9-9445J 
9-94445 
9.94438 
9.94431 



87951 
937 
923 
909 



9.94424 

9-944I7 
9.94410 
9.94404 
9-94397 



854 
840 
826 
87812 
798 
784 
770 
756 



9.94390 
9-94383 
9-94376 
9.94369 
994362 

9-94355 
9-94349 
9.94342 

9-94335 
9.94328 



87743 
729 

7i5 
701 
687 



9.94321 
9.94314 
9-94307 
9.94300 
9.94293 



87673 
659 
645 
631 
617 



9.94286 
9.94279 

9-94273 
9.94266 
9.94259 



87603 
589 
575 
56i 
546 



9.94252 
9.94245 
9.94238 
9.94231 
9.94224 



87532 
5i8 
504 
490 
476 
462 



9.94217 
9.94210 
9.94203 
9.94196 
9.94189 
9.94182 



Nat.TanLog. c.d. Log. Cot Nat. 



53171 
208 
246 
283 
320 



9.72567 
9.72598 
9.72628 
9.72659 
9.72689 



53358 
395 
432 
470 

507 



9.72720 
9.72750 
9.72780 
9.7281 1 
9.72841 



53545 
582 
620 

657 
694 



9.72872 
9.72902 
9.72932 
9.72963 
9-72993 



53732 
769 
807 

844 

882 



9-73023 
9-73054 
9.83084 
9-73i 14 
9-73144 



5392o 
957 
995 

54032 
070 



9-73175 
9-73205 
9-73235 
9-73265 
9-73295 



54107 
145 
183 
220 
258 



9-73326 
9-73356 
9-73386 
9.73416 
973446 



54296 
333 
37i 
409 
446 



9-73476 
9-73507 
9-73537 
9-73567 
973597 



54484 
522 
560 
597 
635 



9.73627 

9-73657 
9.73687 

9-737I7 
9-73747 



54073 
711 
748 
786 
824 



9-73777 

9.73807 

9-73837 
9.73867 

973897 



54862 
900 
938 
975 

55oi3 



9-73927 
9-73957 
973987 
9.74017 
9.74047 



55051 
089 
127 

165 
203 



9.74077 
9.74107 

9-74137 
9.74166 
9.74196 



55241 
279 

317 
355 
393 
43i 



9.74226 
9.74256 
9.74286 
9.74316 
9-74345 
9-74375 



0.27433 
0.27402 
0.27372 
0.27341 
0.2731 1 



1.8807 

794 
781 
768 
755 



0.27280 
0.27250 
0.27220 
0.27189 
0-27I59 



1.8741 
728 

7i5 
702 



0.27128 
0.27098 
0.27068 
0.27037 
0.27007 



1.8676 
663 
650 

637 

624 



0.26977 
0.26946 
0.26916 
0.26886 
0.26856 



1.8611 
598 
585 
572 
559 



0.26825 
0.26795 
0.26765 
0.26735 
0.26705 



1.8546 

533 
520 
507 
495 



0.26674 
0.26644 
0.26614 
0.26584 
0.26554 



3482 
469 
456 
443 
43° 



0.26524 
0.26493 
0.26463 
0.26433 
0.26403 



1.8418 
405 
392 
379 
3°7 



0.26373 
0.26343 
0.26313 
0.26283 
0.26253 



1.8354 
34i 
329 
316 
303 



0.26223 
0.26193 
0.26163 
0.26133 
0.26103 



3291 
278 
265 

253 

240 



0.26073 
0.26043 
0.26013 
0.25983 
0.25953 



1.8228 

215 
202 
190 
177 



0.25923 
0.25893 
0.25863 
0.25834 
0.25804 



S165 

I 5 2 
140 
127 
"5 



0.25774 
0.25744 
0.25714 
0.25684 
0.25655 
0.25625 



1.8103 
090 
078 
065 

053 
040 



Nat. COS Log. d. 



Nat. Sin Log. d. 



Nat. Cot Log. c.d. Log.TanNat 



61 









< 


29 











; 


Nat. Sin Log. d. 


Nat. COS Log. 


d. 


Nat Tan Log. 


c.d. 


Log. Cot Nat. 







48481 9.68557 


23 


874b2 9.94182 


7 


55431 9.74375 


30 
30 
3° 

29 
30 
30 
29 


0.25625 1.8040 


60 


i 


50b 9.68580 


448 9.94175 


469 9.74405 


0-25595 °28 


59 


2 


532 9.68603 


23 


434 9-94168 


7 
7 


507 9.74435 


0.25565 016 


58 


3 


557 9.68625 




420 9.9416 1 


545 9.74465 


0.25535 o°3 


57 


4 
5 


583 9.68648 


"3 
23 


406 9.94154 


7 

7 


583 9-74494 


0.25506 1.7991 
O.25476 1.7979 


56 
55 


48bo8 9.68671 


87391 9-94147 


55621 9-74524 


6 


634 9.68694 


23 


377 9-94I40 


7 
7 


659 9-74554 


0.25446 96b 


54 


7 


6^9 9.68716 




363 9-94133 


697 9-74583 


0.25417 954 


53 


8 


684 9.68739 


2 3 
23 
22 


349 9.94126 


7 

7 
7 
7 


736 9.74613 


3° 
3° 
30 


0.25387 942 


52 


9 


710 9.68762 


335 9.941 19 
87321 9.94112 


774 974643 


0.25357 930 
0.25327 1.7917 


51 
50 


10 


4 8 735 9-68784 


55812 9.74673 


ii 


761 9.68807 


23 


30b 9.94105 


850 9.74702 


29 


0.25298 905 


49 


12 


786 9.68829 




292 9.94098 


7 
8 


888 9.74732 


3° 


0.25268 893 


48 


13 


811 9.68852 


2 3 


278 9.94090 




926 9.74762 


3° 


0.25238 881 


47 


14 


837 9.68875 


23 
22 


2b4 9.94083 
87250 9.94076 


7 
7 


964 9-74791 


29 
30 


0.25209 868 


4 b 


15 


488b2 9.68897 


5boo3 9.74821 


O.25179 1.785b 


45 


ib 


888 9.68920 


23 


235 9-94069 


7 
7 
7 


041 9.74851 


3° 
29 
30 


0.25149 844 


44 


17 


913 9.68942 


23 


221 9.94062 


079 9.74880 


O.25120 832 


43 


18 


938 9.68965 


207 9-94055 


117 9.74910 


0.25090 820 


4 2 


19 


9b4 9.68987 


23 


193 9.94048 


7 
7 
7 
7 


156 9.74939 


29 
30 
29 
30 


0.25061 808 


4i 


20 


48989 9.69010 


87178 9.94041 


56194 9.74969 


0.25031 1.7796 


40 


21 


49014 9.69032 


23 


164 9.94034 


232 9.74998 


0.25002 783 


39 


22 


040 9.69055 


150 9.94027 


270 9.75028 


0.24972 771 


38 


23 


065 9.69077 


23 
22 


136 9.94020 


7 
8 


309 9.75058 


3° 
29 
30 
29 


0.24942 759 


37 


24 


090 9.69100 


121 9.94012 


7 

7 


347 9.75087 


0.24913 747 


3^ 


25 


491 16 9.69122 


87107 9.94005 


56385 9.75"7 


0.24883 1.7735 


35 


2b 


141 9.69144 


~~ 


093 9-93998 


424 9-75*46 


0.24854 723 


34 


27 


ibb 9.69167 


2 3 


079 9-93991 


7 


4b2 9.75176 


3° 


0.24824 711 


33 


28 


192 9.69189 


23 
22 


ob 4 9.93984 


7 
7 
7 


501 9.75205 


29 
30 
29 


0.24795 699 


32 


29 


217 9.69212 


050 9-93977 


539 9.75235 


0.24765 687 


3i 
30 


30 


49242 9.69234 


8703b 9.93970 


56577 975264 


0.24736 1.7675 


3i 


268 9.69256 


23 


021 9.93963 


7 
8 


616 975294 


3° 
29 


O.24706 6b3 


29 


32 


293 9.69279 


007 9-93955 




654 975323 


0.24677 651 


28 


33 


318 9.69301 


00 


86993 9-93948 


7 


693 9-75353 


3° 


O.24647 639 


27 


34 
35 


344 9-69323 


22 
23 


978 9-93941 


7 
7 

7 
7 
8 

7 
7 


73i 9-75382 


29 

29 

3° 
29 
30 
29 
29 


O.24618 627 


2b 

25 


49369 9-69345 


86964 9-93934 


56769 9-7541 1 


O.24589 1.7615 


30 


394 9-69368 


949 9-93927 


808 9.75441 


O.24559 bo 3 


24 


37 
38 


419 9-69390 
445 9-694I2 


22 


935 9.93920 
921 9.93912 


846 9-75470 
885 9-75500 


O.24530 591 
0.24500 579 


23 
22 


39 
40 


470 9.69434 


22 
23 


906 9-93905 


923 975529 


0.24471 5b7 


21 
20 


49495 9-69456 


8b892 9.93898 


5b 9 b2 9.75558 


0.24442 1.755b 


41 


521 9-69479 


878 9.93891 


7 


57000 9.75588 


3° 


0.24412 544 


19 


42 


54b 9.69501 


00 


8b 3 9.93884 


7 
8 


o39 9-756I7 


29 
30 
29 
29 
30 


0.24383 532 


18 


43 


571 9-69523 


00 


849 9-93876 


7 
7 


078 9.75647 


0.24353 520 


17 


44 


596 9-69545 


22 


834 9-93869 


116 9.75676 


0.24324 508 


ib 


45 


49b22 9.69567 


86820 9.93862 


57155 9-75705 


0.24295 1.749b 


15 


4 b 


647 9-69589 


on 


805 9-93855 


7 
8 


193 9-75735 


0.24265 485 


14 


47 


672 9.6961 1 


no 


791 9.93847 




232 9-75764 


29 


0.24236 473 


13 


48 


697 9.69633 


on 


777 9-93840 


7 


271 9-75793 


29 


0.24207 461 


12 


49 


723 9.69655 


22 


762 9.93833 


7 
7 


309 9.75822 


29 
30 


0.24178 449 


11 


50 


49748 9.69677 


8b748 9.93826 


57348 9.75852 


0.24148 1.7437 


10 


51 


773 9-69699 


no 


733 9-938I9 


7 
8 


386 9.75881 :* 0.241 19 4 2b 


9 


52 


798 9.69721 


on 


719 9.9381 1 




425 9.75910 




0.24090 414 


8 


.S3 


824 9.69743 


~~ 


704 9.93804 


7 


464 9-75939 


29 


0.24061 402 


7 


54 


849 9.69765 


22 


690 9-93797 


7 
8 


503 9-75969 


3° 
29 


0.24031 391 


b 
5 


55 


49874 9-69787 


86675 9.93789 


57541 9.75998 


0.24002 1.7379 


56 


899 9.69809 


nn 


6b 1 9.93782 


7 


580 9.76027 


29 


0.23973 367 


4 


57 


924 9.69831 


on 


646 9-93775 


7 


619 9.76056 


29 


0.23944 355 


3 


SB 


950 9-69853 


nn 


632 9-93768 


7 
8 


657 9.76086 


3° 


0.23914 344 


2 


% 


975 9-69875 


nX 


617 9-9376o 


7 


696 9.761 15 


29 

29 


0.23885 332 


1 


BO 


50000 9.69897 




603 9-93753 


735 9-76144 


0.23856 321 







Nat. COS Log. d. 


Nat. Sin Log 


d. 


Nat. Cot Log. c.d. Log.Tan Nat. 


/ 



60 c 









30 


O 








f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







50000 9.69897 




86603 9-93753 




57735 9-76I44 




0.23856 1.7321 


60 


I 


025 9.69919 


22 


588 9.93746 


7 

8 


774 9.76173 


29 


0.23827 309 


S9 


2 


050 9.69941 


22 


573 9-93738 




813 9.76202 


29 


0.23798 297 


S8 


3 


076 9.69963 




559 9-93731 


7 


851 9.76231 


29 


0.23769 286 


S7 


4 
5 


101 9.69984 


22 


544 9-937 2 4 


7 

7 
3 


890 9.76261 


3° 
29 
29 


0.23739 274 


56 
55 


50126 9.70006 


86530 9-93717 


57929 9.76290 


0.23710 1.7262 


6 


151 9.70028 




5i5 9-93709 




968 9.76319 


0.23681 251 


S4 


7 


176 9.70050 




501 9.93702 


7 

7 

8 


58007 9.76348 


29 
29 
29 
29 

29 
29 
29 
29 
29 
29 


0.23652 239 


S3 


8 


201 9.70072 


21 


486 9.93695 


046 9.76377 


0.23623 228 


S2 


9 


227 9.70093 


22 


471 9.93657 


7 
7 

8 


085 9.76406 


0.23594 216 


51 
50 


10 


50252 9.70115 


86457 9.93680 


58124 9.76435 


0-23565 I -7205 


ii 


277 9-70I37 


22 


442 9-93673 


162 9.76464 


0.23530 193 


49 


12 


302 9.70159 




427 9.93665 




201 9.76493 


0.23507 182 


48 


13 


327 9.70180 




4i3 9-93658 


7 

8 


240 9.76522 


0.23478 170 


47 


14 


352 9.70202 


22 


398 9.93650 


7 


279 9-7655I 


0.23449 159 


46 
45 


15 


50377 9.70224 


86384 9.93643 


58318 9.76580 


0.23420 1.7147 


16 


403 9.76245 




369 9.93636 


7 
8 


357 9-76609 


0.23391 136 


44 


17 


428 9.70267 




354 9.93628 


396 9.76639 


3° 
29 
29 
28 
29 
29 
29 
29 
29 

29 
29 
29 
29 
29 
29 


0.23361 124 


43 


18 


453 9-7 0288 




340 9.93621 


7 
7 
8 

7 

8 


435 9-76668 


0.23332 113 


42 


19 


478 9.70310 


22 


325 9-936I4 


474 9-76697 


0.23303 102 


41 


20 


50503 9.70332 


86310 9.93606 


58513 9.76725 


0.232^5 1.7090 


40 


21 


528 9-70353 




295 9-93599 


552 9-76754 


0.23246 079 


39 


22 


553 9-70375 




28 1 9-93591 




591 9.76783 


0.23217 067 


.38 


23 


578 9.70396 




266 9.93584 


7 
7 
8 


631 9.76812 


0.23188 056 


37 


24 


603 9.70418 


21 


251 9-93577 


670 9.76841 


0.23159 045 


30 


25 


50628 9.70439 


86237 9.93569 


58709 9.76870 


0.23130 1.7033 


35 


26 


654 9.70461 




222 9.93562 


7 
8 


748 9-76899 


0.23101 022 


.34 


27 


679 9.70482 




207 9-93554 


7 
8 


787 9.76928 


0.23072 on 


33 


28 


704 9.70504 


21 


192 9-93547 


826 9.76957 


0.23043 1.6999 


.32 


29 


729 9-705 2 5 


22 


178 9-93539 


7 

7 
8 


865 9.76986 


0.23014 988 


3i 
30 


30 


50754 9.70547 


86163 9-93532 


58905 9.77015 


0.229^5 1.6977 


31 


779 9-70568 




148 9.93525 


944 9-77044 


0.22956 905 


29 


32 


804 9.70590 




133 9-935!7 




983 9.77073 2 8 


0.22927 954 


28 


33 


829 9.7061 1 


22 


119 9-93510 


7 

8 


59022 9.77101 ; ori 


0.22899 943 


27 


34 


854 9-70633 


21 


104 9.93502 


7 
8 


061 9.77130 


29 


O.22870 932 


26 
25 


35 


50879 9.70654 


86089 9.93495 


59101 9.77159 


0.22841 1.6920 


36 


904 9.70675 




o74 9.93487 




140 9.77188 | * 


0.22812 909 


24 


37 


929 9.70697 




o59 9-9348o 


7 

8 


179 9-772I7 


29 

28 


0.22783 898 


23 


38 


954 9-707I8 




045 9-93472 


7 
8 


218 9.77246 


0.22754 887 


22 


39 
40 


979 9-70739 


22 


030 9-93465 


258 9-77274 


29 
29 

29 
29 
~8 


0.22726 875 


21 
20 


51004 9.70761 


86015 9-93457 


59297 9-77303 


0.22697 1-6864 


41 


029 9.70782 




000 9.93450 


7 
g 


336 9-77332 


0.22668 853 


19 


42 


054 9.70803 




85985 9-93442 




376 9-7736i 


0.22639 842 


18 


43 


079 9.70824 


22 


97o 9-93435 


8 


4i5 9-77390 


0.22610 831 


17 


44 


104 9.70846 


21 


956 9.93427 


7 
8 


454 9-77418 


29 
29 
29 

"8 


0.22582 820 


ib 
"15 


45 


51 129 9.70^67 


85941 9.93420 


59494 9-77447 


0.22553 1.6808 


46 


154 9.70888 




926 9.93412 


7 
8 


533 9-77476 


0.22524 797 


14 


47 


179 9.70909 




911 9.93405 


573 977505 


0.22495 786 


13 


48 


204 9.70931 




896 9-93397 


7 
8 

7 

8 


612 9.77533 


29 
29 


0.22467 775 


12 


49 
"50 


229 9.709^2 


21 


881 9.93390 


651 9-77562 


0.22438 764 


n 


51254 9.70973 


85866 9.93382 


59691 9.77591 


0.22409 1.6753 


10 


Si 


279 9-70994 




851 9-93375 


730 9.77619 


29 
29 
29 
28 
29 
08 


0.22381 742 


9 


S2 


304 9-71015 




836 9.93367 


7 
3 


770 9.77648 


0.22352 731 


8 


S3 


329 9.71036 


22 


821 9.93360 


809 9.77677 


0.22323 720 


7 


54 
55 


354 9-7I058 


21 


806 9-93352 


8 

7 

3 


849 9.77706 


0.22294 709 


b 


51379 9.71079 


85792 9-93344 


59888 9.77734 


0.22266 1.6698 


5 


S6 


404 9.71 100 


21 


777 9-93337 


928 9.77763 


0.22237 687 


4 


S7 


429 9.71121 




762 993329 




967 9-77791 


29 
29 

28 


0.22209 676 


3 


S8 


454 9-7II42 




747 9-93322 


7 
8 


60007 9.77820 


0.22180 665 


2 


u 


479 9-7ii63 




732 9.93314 




046 9.77849 


0.22151 654 


1 


504 9.71184 




717 9-93307 


7 


086 9.77877 




0.22123 643 | 




Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. jc.d.| Log. Tan Nat. 


' 



59 









31° 








f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







51504 9.71184 




85717 9-93307 


8 


60086 9.77877 


29 
29 

28 


0.22123 1.6643 


60 


I 


529 9.71205 




702 9.93299 


8 


126 9.77906 


0.22094 632 


59 


2 


554 9-71226 




687 9.93291 


7 
8 


165 9-77935 


0.22065 621 


58 


3 


579 9-7I247 




672 9.93284 


205 9-77963 




0.22037 610 


57 


4 
5 


604 9.71268 


21 


6 57 9-93 2 76 


7 
8 


245 9-77992 


29 
28 
29 
28 


0.22008 599 


50 
55 


51628 9.71289 


85642 9.93269 


60284 9.78020 


0.21980 1.6588 


6 


653 9-7i3io 




627 9.93261 


8 


324 9.78049 


0.21951 577 


54 


7 


678 9-7I33 1 




612 9.93253 


7 

8 


364 9.78077 


29 
29 

28 

29 

28 


0.21923 566 


53 


8 


703 9-7I352 




597 9-93246 


403 9.78106 


0.21894 555 


52 


9 


728 9.71373 


20 


582 9.93238 


8 

7 
8 


443 9-78I35 


0.21865 545 


5i 
50 


10 


51753 9-7I393 


85567 9-93230 


60483 9.78163 


0.21837 1.6534 


ii 


778 9.71414 




551 9-93223 


522 9.78192 


0.21808 523 


49 


12 


803 9.71435 




536 9.93215 


8 


562 9.78220 


29 
28 


0.21780 512 


48 


13 


828 9.71456 




521 9.93207 


7 
8 

8 


602 9.78249 


0.21751 501 


47 


14 

15 


852 9.71477 


21 


506 9.93200 


642 9.78277 


29 


0.21723 490 


40 
45 


51877 9.71498 


85491 9-93I92 


606b 1 9.78306 


0.21694 1.6479 


ib 


902 9.71519 




476 9.93184 




721 9.78334 


29 

28 


0.21666 469 


44 


17 


927 9-7!539 




461 9.93177 


7 
8 
8 


761 9.78363 


0.21637 458 


43 


18 


952 9.71560 




446 9.93169 


801 9.78391 


28 


0.21609 447 


42 


19 
"20 


977 9-7i58i 


21 


431 9.93161 


7 
8 


841 9.78419 


29 

28 


0.21581 436 


4i 


52002 9.71602 


85416 9-93I54 


60881 9.78448 


0.21552 1.6426 


40 


21 


026 9.71622 




401 9.93146 


8 


921 9.78476 


29 


0.21524 415 


39 


22 


051 9.71643 




385 9-93I38 




960 9.78505 


0.21495 404 


38 


23 


076 9.71664 




370 9.93131 


7 

8 


61000 9.78533 


29 
28 

28 


0.21467 393 


37 


24 
25 


101 9.71685 


20 


355 9-93I23 


8 


040 9.78562 


0.21438 383 


30 


52126 9.71705 


85340 9-93"5 


bioSo 9.78590 


0.21410 1.6372 


35 


26 


151 9.71726 




325 9-93io8 


7 


120 9.78618 


29 

^8. 


0.21382 361 


34 


27 


175 9-7I747 




310 9.93100 


8 


160 9.78647 


0.21353 351 


33 


28 


200 9.71767 


£.i. 


294 9.93092 


8 


200 9.78675 


29 
28 
28 


0.21325 340 


32 


29 


225 9.71788 


21 


279 9-93084 


7 
8 


240 9.78704 


0.21296 329 


3i 


30 


52250 9.71809 


85264 9.93077 


61280 9.78732 


0.21268 1.63 19 


30 


31 


275 9.71829 


' L 


249 9-93069 


8 
8 


320 9.78760 


29 

28 


0.21240 308 


29 


32 


299 9-71850 




234 9-93o6i 


360 9.78789 


0.21211 297 


28 


33 


324 9.71870 




218 9.93053 




400 9.78817 


28 


0.21183 287 


27 


34 
35 


349 9-71891 


20 


203 9.93046 


7 
8 
8 


440 9.78845 


29 

28 


0.21 155 276 


2b 


52374 9-7I9II 


85188 9.93038 


61480 9.78874 


0.21126 1.6265 


25 


30 


399 9-7I932 




173 9-93030 


% 


520 9.78902 


^8 


0.21098 255 


24 


37 


423 9-7I952 




157 9.93022 


8 


561 9.78930 


29 

~8 


0.21070 244 


23 


3* 


448 9.71973 




142 9.93014 


7 
8 
9 


601 9.78959 


0.21041 234 


22 


39 
40 


473 9-7I994 


20 


127 9.93007 


641 9.78987 


28 
"8 


0.21013 223 


21 


52498 9.72014 


85112 9.92999 


61681 9.79015 


0.20985 1.6212 


20 


41 


522 9.72034 




096 9.92991 


8 


721 9.79043 


29 

28 


0.20957 202 


19 


42 


547 9-72055 




081 9.92983 




7bi 9.79072 


0.20928 191 


18 


43 


572 9-72075 




066 9.92976 


7 
8 
8 
8 


801 9.79100 


28 


0.20900 181 


17 


44 
45 


597 9-72096 


20 


051 9.92968 


842 9.79128 


28 

29 

08 


0.20872 170 


ib 


52621 9.72116 


85035 9-92960 


bi882 9.79156 


0.20844 1.6160 


15 


4 b 


646 9-72I37 




020 9.92952 


g 


922 9.79185 


0.20815 149 


14 


47 


671 9.72157 




005 9-9 2 944 


Q 


962 9.79213 


28 


0.20787 139 


13 


48 


696 9.72177 




84989 9.92936 


U 

7 
8 
8 


62003 9-7924I 


^0 


0.20759 128 


12 


49 


720 9.72198 


20 


974 9-9 2 9 2 9 


043 9-79269 


28 
29 

28 


0.20731 118 


11 


50 


52745 9.72218 


84959 9-9292I 


62083 9.79297 


0.20703 1.6107 


10 


5i 


770 9.72238 




943 9-929 J 3 


« 


124 9.79326 


0.20674 097 


9 


52 


794 9-72259 




928 9.92905 


8 


164 9-79354 


28 


0.20646 087 


8 


53 


819 9.72279 


4KJ 


913 9-92897 


g 


204 9.79382 


28 


0.20618 076 


7 


54 
55^ 


844 9.72299 


21 


897 9.92889 


8 


245 9.79410 


28 
28 


0.20590 066 


b 
5 


52869 9.72320 


84882 9.92881 


62285 9.79438 


0.20562 1.6055 


so 


893 9-72340 




866 9.92874 


7 
8 


325 9-79466 




0.20534 045 


4 


57 


918 9.72360 




851 9.92866 


8 


366 9.79495 


29 
28 


0.20505 034 


3 


5* 


943 9-7238I 




836 9.92858 


8 


406 9.79523 


^8 


0.20477 024 


2 


59 


967 9.72401 




820 9.92850 


8 


446 9-7955 1 


28 


0.20449 014 


1 


60 


992 9.72421 




805 9.92842 




487 979579 




0.20421 003 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. c.d. 


Log. Tan Nat. 


f 



58 c 



32° 



t 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. c.d. 


Log. Cot Nat. 




5 2 992 9.72421 




84805 9.92842 


3 


62487 9.79579 


28 


0.20421 1.6003 


60 


I 


53017 972441 




789 9.92834 


8 


527 9.79607 


28 


0.20393 1.5993 


59 


2 


041 9.72461 




774 9.92826 


8 


568 9.79635 


28 


0.20365 983 


58 


3 


ob6 9.72482 




759 9.92818 


8 


608 9.79663 


28 


0.20337 972 


57 


4 
5 


091 9.72502 


20 


743 9.92810 


7 

8 


649 9.79691 


28 
"8 


0.20309 962 


56 
55 


53 1 15 9.72522 


84728 9.92803 


62689 9.79719 


0.20281 1.5952 


b 


140 9.72542 




712 9.92795 


8 


730 9.79747 


29 

28 


0.20253 941 


54 


7 


164 9.72562 




697 9.92787 


8 


770 9.79776 


0.20224 931 


53 


8 


189 9.72582 




681 9.92779 


8 


811 9.79804 


28 


0.20196 921 


52 


9 


214 9.72602 


20 


666 9.92771 


8 

8 


852 9-79832 


28 
28 


0.20168 911 


5i 
50 


10 


53238 9.72622 


84650 9.92763 


62892 9.79860 


0.20140 1.5900 


11 


2b3 9.72643 




635 9-92755 


8 


933 9.79888 


"8 


0.20112 890 


49 


12 


288 9.72663 




619 9.92747 


9 


973 9-79916 


o^ 


0.20084 880 


48 


13 


312 9.72683 




604 9.92739 


8 


63014 9.79944 


^8 


0.20056 869 


47 


14 


337 9-72703 


20 


588 9-9273I 


8 
8 


o55 9-79972 


28 

"8 


0.20028 859 


40 
45 


15 


53361 9.72723 


84573 9-92723 


63095 9.80000 


0.20000 1.5849 


lb 


386 9-72743 




557 9-927I5 


8 


136 9.80028 


-8 


0.19972 839 


44 


17 


411 9-72763 




542 992707 


g 


177 9.80056 


28 


0.19944 829 


43 


18 


435 9-72783 


~ 


526 9.92699 


8 


217 9.80084 


09 


0.19916 818 


42 


19 


460 9.72803 


20 


511 9.92691 


8 

3 


258 9.80112 


28 

~8 


0.19888 808 


4i 
40 


20 


53484 9.72823 


84495 9-92683 


63299 9.80140 


0.19860 1.5798 


21 


509 9.72843 




480 9.92675 


8 


340 9.80168 


27 

28 


0.19832 788 


30 


22 


534 9.72863 




464 9.92667 


g 


380 9.80195 


0.19805 778 


38 


23 


558 9-72883 


19 
20 


448 9.92659 


8 


421 9.80223 


28 


0.19777 768 


37 


24 


583 9.72902 


433 9-9265I 


8 
8 


462 9.80251 


28 
"8 


0.19749 757 


36 
35 


25 


53607 9.72922 


84417 9.92643 


63503 9-80279 


0.19721 1.5747 


2b 


632 9.72942 




402 9.92635 


8 


544 9.80307 


2S 


0.19693 737 


34 


27 


656 9.72962 




386 9.92627 


8 


584 9.80335 


28 


0.19665 727 


33 


28 


681 9.72982 




370 9.92619 


R 


625 9.80363 


^8 


0.19637 717 


32 


29 


705 9-73002 


20 
19 


355 9-926ii 


8 

8 


666 9.80391 


28 
"8 


0.19609 707 


3 1 
30 


30 


5373° 9-73022 


84339 9-92603 


63707 9.80419 


0.19581 1.5697 


31 


754 9-7304I 


324 9.92595 


8 


748 9-80447 


27 
28 


0.19553 687 


29 


32 


779 9-73o6i 


" 


308 9.92587 


8 


789 9.80474 


0.19526 677 


28 


33 


804 9.73081 




292 9.92579 


8 


830 9.80502 


^8 


0.19498 667 


27 


34 
35 


828 9.73101 


20 

19 


277 9-9257I 


8 
8 


871 9.80530 


28 

"8 


0.19470 657 


2b 
25 


53853 9-73i2i 


84261 9.92563 


63912 9.80558 


0.19442 1.5647 


3* 


877 9-73 x 40 


245 992555 


9 

8 


953 9-80586 


^8 


0.19414 637 


24 


37 


902 9.73160 




230 992546 


994 9.80614 


28 


0.19386 627 


23 


38 


926 9.73180 


" 


214 9.92538 


8 


64035 9.80642 


27 
28 


0.19358 617 


22 


39 
40 


951 9.73200 


19 


198 9-92530 


8 

8 


076 9.80669 


0.19331 607 


21 
20 


53975 9-73219 


84182 9.92522 


641 17 9.80697 


0.19303 1.5597 


41 


54000 9.73239 




167 9.92514 


8 


158 9.80725 


oR 


0.19275 587 


19 


42 


024 9.73259 


19 


151 9.92506 


R 


199 9.80753 


oR 


0.19247 577 


18 


43 


049 9.73278 


135 9.92498 


8 


240 9.80781 


27 
28 
28 


0.19219 567 


17 


44 


073 9-73298 


20 


120 9.92490 


8 

9 

8 


281 9.80808 


0.19192 557 


ib 
15 


45 


54097 9-73318 


84104 9.92482 


64322 9.80836 


0.19164 1.5547 


46 


122 9.73337 


I 9 


088 9.92473 


363 9.80864 


"8 


0.19136 537 


14 


47 


146 9-73357 




072 9.92465 


8 


404 9.80892 


27 

28 


0.19108 527 


13 


48 


171 9-73377 


19 
20 

19 


057 9.92457 


8 


446 9.80919 


0.19081 517 


12 


49 
50 


195 9-73396 


041 9.92449 


8 

8 


487 9.80947 


28 
28 


0.19053 507 


11 


54220 9.73416 


84025 9.92441 


64528 9.80975 


0.19025 1.5497 


10 


51 


244 9-73435 


009 9.92433 


8 


569 9.81003 


27 

28 


0.18997 487 


9 


S2 


269 9-73455 


t" 


83994 9-92425 


9 

8 


610 9.81030 


0.18970 477 


8 


S3 


293 9-73474 


I 9 


978 9.92416 


652 9.81058 


28 


0.18942 468 


7 


54 


3i7 9-73494 


19 


962 9.92408 


8 

8 


693 9.81086 


27 


0.18914 458 


b 


55 


54342 9.73513 


83946 9.92400 


64734 9.81 1 13 


0.18887 1.S448 


5 


56 


366 9-73533 


** 


93o 9-92392 


8 


775 9.81 141 


28 


0.18859 438 


4 


57 


39i 9-73552 


I 9 


915 9.92384 


8 


817 9.81169 


27 

28 


0.18831 428 


3 


SB 


4i5 9-73572 


" 


899 9.92376 


9 
8 


858 9.81 196 


0.18804 418 


2 


S9 


44o 9-73591 


x 9 


883 9.92367 


899 9.81224 


28 


0.18776 408 


1 


60 


464 9.7361 1 




867 9.92359 




941 9.81252 




0.18748 399 







Nat.CoSLog. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


t 



57 











33 











f 


Nat. Sin Log. d. 


Nat. COS Log 


d. 


Nat. Tan Log. c.d. 


Log. Cot Nat. 







54464 9-736ii 


19 


83867 9.92359 


8 


64941 9.81252 1 

982 9.81279 7 


0.18748 1.5399 


60 


I 


488 9.73630 


851 9-92351 


8 


0.18721 389 


59 


2 


5i3 9-73650 


19 


835 9-92343 


8 


65024 9.81307 8 


0.18693 379 


58 


3 


537 9-73669 


819 9-92335 


9 

8 

8 


065 9.81335 J; ! 0.18665 369 


57 


4 


561 9.73689 


19 
19 


804 9.92326 


106 9.81362 


28 


0.18638 359 


56 


5 


54586 9-73708 


83788 9.92318 


65148 9.81390 


0.18610 1.5350 


55 


b 


610 9.73727 


772 9.92310 


8 


189 9.81418 ~: j 0.18582 340 
231 9.81445 / 0.18555 330 


54 


7 


635 9-73747 


19 
19 
20 

19 


756 9.92302 


9 

8 


53 


8 


659 9-73766 


740 9-92293 


272 9-81473 „ 


0.18527 320 


52 


9 


683 9-73785 


724 9.92285 


8 
8 


314 9.81500 


28 

08 


0.18500 311 
0.18472 1.5301 


5i 
50 


10 


54708 9-73805 


83708 9.92277 


65355 9.81528 


ii 


732 9.73824 


692 9.92269 




397 9-8I556 27 


0.18444 291 


49 


12 


756 9.73843 


19 


676 9.92260 


9 

8 


438 9-8I583 1 


0.18417 282 


48 


13 


781 9.73863 




660 9.92252 


8 


480 9.81611 : ; 


0.18389 272 


47 


14 


805 9.73882 


I 9 
19 


645 9.92244 


9 
8 


521 9.81638 


28 


0.18362 262 


46 
45 


15 


54829 9.73901 


83629 9.92235 


65563 9.81666 


0.18334 1.5253 


lb 


854 9-73921 


19 
19 
19 
19 


613 9.92227 


8 


604 9.81693 ;' 0.18307 243 


44 


17 


878 9-73940 


597 9.92219 


8 


646 9.8I72I ; f 


0.18279 233 


43 


18 


902 9.73959 


581 9.9221 1 




688 9.81748 % 


0.18252 224 


42 


19 


927 9-73978 


565 9.92202 
83549 9-92194 


9 
8 
8 


729 9.81776 


27 


0.18224 214 
0.18197 1.5204 


4i 
40 


20 


54951 9-73997 


65771 9.81803 


21 


975 9-74017 




533 9-92186 


813 9.81831 1 ~ 


0.18169 195 


39 


22 


999 9-74036 


19 
19 
19 
19 


517 9-92177 


9 
8 
8 

9 
8 


854 9.81858 % [ 0.18142 185 


38 


23 


55024 9.74055 


501 9.92169 


896 9.81886 ^ \ 0.18114 175 


37 


24 


048 9-74074 


485 9.92161 


938 9.81913 ^7 
65980 9.81941 i 


0.18087 166 


36 


25 


55072 9.74093 


83469 9.92152 


0.18059 1.515b 


35 


2b 


097 9-74II3 


19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 
19 


453 9-92I44 


8 


06021 9.81968 L 0.18032 147 


34 


27 


121 9.74132 


437 9-92I36 




063 9.81996 < f 


0.18004 137 


33 


28 


145 9-74I5I 


421 9.92127 


9 


105 9.82023 j; 


0.17977 127 


32 


29 


169 9.74170 


405 9.921 19 


8 


147 9.82051 


08 


0.17949 118 


3i 


30 


55194 9.74189 


833 8 9 9-92IH 


66189 9.82078 


0.17922 1.5108 


30 


31 


218 9.74208 


373 9.92102 


9 
8 

8 


230 9.82106 ~ 


0.17894 099 


29 


32 


242 9-74227 


356 9.92094 


272 9.82133 j % 0.17867 089 


28 


33 


266 9.74246 


340 9.92086 


314 9.82161 




0.17839 080 


27 


34 


291 9.74265 


324 9.92077 


9 
8 


356 9.82188 


27 

27 

28 


0.17812 070 


2b 

25 


35 


55315 9.74284 


83308 9.92069 


66398 9.82215 


0.17785 1. 506 1 


3^ 


339 9-74303 


292 9.92060 


9 

8 


440 9.82243 


27 

28 


0.17757 051 


24 


37 


363 9.74322 


276 9.92052 


8 


482 9.82270 


0.17730 042 


23 


3» 


388 9-74341 


260 9.92044 


524 9.82298 




0.17702 032 


22 


39 
40 


4 12 9-74360 


I 9 
19 


244 9.92035 


9 

8 


566 9.82325 


27 

27 
28 
27 

28 


0.17675 023 


21 
20 


55436 9.74379 


83228 9.92027 


66608 9.82352 


0.17648 1.5013 


41 


460 9.74398 


I 9 
19 
19 


212 9.92018 


9 
8 


650 9.82380 


0.17620 004 


19 


42 


484 9-744I7 


195 9.92010 


8 


692 9.82407 


0.17593 1.4994 


18 


43 


509 9.74436 


179 9.92002 


734 9-82435 




0.17565 985 


17 


44 


533 9-74455 


I 9 

19 
19 
19 
19 

18 


163 9-9I993 


9 
8 


776 9.82462 


27 


O.I7538 975 


ib 


45 


55557 9-74474 


83147 9-91985 


66818 9.82489 


27 
08 


0.17511 1.4966 


15 


4 b 


581 9.74493 


131 9.91976 


9 

8 

9 
8 


860 9.82517 ~ j 0.17483 957 

902 9.82544 ! / j 0.17456 947 
944 9.82571 { o £ | 0.17429 938 


14 


47 


605 9-745 12 


115 9.91968 


13 


48 


630 9-74531 


098 9.91959 


12 


49 


654 9-74549 


19 
19 
19 
19 


082 9.91951 


9 
8 


986 9.82599 


1 0.17401 928 

27 — — — 

' 0.17374 14919 


11 

To 


50 


55678 9.74568 


83066 9.91942 


67028 9.82626 


5i 


702 9.74587 


050 9.91934 


9 
8 


071 9.82653 -^ 0.17347 910 


9 


52 


726 9.74606 


034 9.91925 


113 9.82681 


27 


0.17319 900 


8 


S3 


750 9-74625 


017 9.91917 




155 9.82708 


0.17292 891 


7 


54 
55 


775 9-74644 


z 9 

18 

19 


001 9.91908 


9 
8 


197 9.82735 


27 

27 

28 


0.17265 882 


b 


55799 9-74662 


82985 9.91900 


67239 9.82762 


0.17238 14872 


5 





823 9.74681 


969 9.91891 


9 

8 


282 9.82790 




0.17210 863 


4 


57 


847 9-74700 


x 9 


953 9-91883 


324 9.82817 


27 


0.17183 854 


3 


58 


871 9-74719 


19 
18 


936 9-91874 


9 
8 


366 9.82844 


27 

27 
28 


0.17156 844 


2 


W 


895 9-74737 




920 9.91866 




409 9.82871 


0.17129 835 


1 


60 


919 9-74756 


x 9 


904 9-9I857 


9 


451 9.82899 


0.17101 826 







Nat. COS Log. d. 


Nat. Sin Log 


d. 


Nat. Cot Log. 


C.d. 


Log.TanNat. 


/ 



56 









34 


O 








f 


Nat. Sin Log. d. 


Nat. COS Log 


d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 




55919 9-74756 


19 
19 
18 


82904 9.91857 


8 


67451 9.82899 


27 
27 

27 

°8 


0.17101 1.4826 


60 


I 


943 9-74775 


887 9.91849 


9 

8 


493 9.82926 


0.17074 816 


S9 


2 


968 9.74794 


871 9.91840 


536 9.82953 


0.17047 807 


S8 


3 


992 9.74812 


19 
19 
18 


855 9-9!832 


9 
8 

9 

8 


578 9.82980 


0.17020 798 


S7 


4 


56016 9.74831 


839 9.91823 


620 9.83008 


27 
27 
27 


0.16992 788 


56 

55" 


5 


56040 9.74850 


82822 9.91815 


67663 9.83035 


0.16965 1.4779 


6 


064 9.74868 


19 


806 9.91806 


705 9.83062 


0.16938 770 


S4 


7 


088 9.74887 


790 9.91798 


9 
8 


748 9.83089 


0.16911 761 


S3 


8 


112 9.74906 


I 9 
18 


773 9-9 J 789 


790 9.831 17 


27 

27 
27 
27 
27 

^8 


0.16883 751 


S2 


9 
10 


136 9.74024 


3 


757 9-9i78i 


9 


832 9-83I44 


0.16856 742 


51 

50 


56160 9.74943 


82741 9.91772 


67875 9-83171 


0.16829 1.4733 


11 


184 9.74961 


19 
19 
18 


724 9.91763 


9 

8 


917 9.83198 


0.16802 724 


49 


12 


208 9.74980 


708 9.91755 


9 

8 


960 9.83225 


0.16775 715 


48 


13 


232 9-74999 


692 9.91746 


68002 9.83252 


0.16748 705 


47 


14 
15 


2 5 6 9-75 OI 7 
56280 9.75036 


19 
18 


675 9-9I738 


9 

9 

8 


045 9.83280 


27 
27 
27 

27 
27 

27 

28 


0.16720 696 


46 
45 


82659 9.91729 


68088 9.83307 


0.16693 1-4687 


16 


3o5 9-75054 


19 
18 


643 9.91720 


130 9-83334 


0.16666 678 


44 


17 


329 9.75073 


626 9.91712 


9 

g 


173 9-8336I 


0.16639 669 


43 


18 


353 9-7509I 


19 
18 

19 
18 


610 9.91703 


215 9-83388 


0.16612 659 


4-2 


19 


377 9-75IIO 


593 9-91695 


9 

9 

8 


258 9.83415 


0.16585 650 


41 
40 


20 


56401 9.75128 


82577 9.91686 


68301 9.83442 


0.16558 1.4641 


21 


425 9-75I47 


561 9.91677 


343 9-83470 




0.16530 632 


39 


22 


449 9-75*05 


19 

18 


544 9.91669 




386 9.83497 


27 


0.16503 623 


38 


23 


473 9-75I84 


528 9.91660 


9 


429 9.83524 


27 


0.16476 614 


-M 


24 


497 9-75202 


19 
18 


511 9-91651 


9 

8 

9 


47i 9-8355I 


27 
27 


0.16449 605 


36 
35 


25 


56521 9.75221 


82495 9-9 J 643 


68514 9.83578 


0.16422 1.4596 


26 


545 9-75239 




478 9.91634 


557 9-83605 


27 


0.16395 586 


34 


27 


569 9-75258 


I 9 
18 
18 


462 9.91625 


9 
8 


600 9.83632 


27 


0.16368 577 


S3 


28 


593 9-75276 


446 9.91617 


9 

9 

8 


642 9.83659 


27 


0.16341 568 


32 


29 


617 9-75294 


19 
18 


429 9.91608 


685 9.83686 


27 
27 


0.16314 559 


31 
30! 


30 


56641 9.75313 


82413 9-9I599 


68728 9.83713 


0.16287 1.4550 


31 


665 9-75331 




396 9.91591 


9 

9 
8 


771 9.83740 


27 


0.16260 541 


29 


32 


68 9 9-75350 


J 9 
18 


380 9.91582 


814 9.83768 




0.16232 532 


28 


33 


7i3 9-75368 


18 


363 9.91573 


857 9-83795 


27 


0.16205 523 


27 


34 


736 9-75386 


19 
18 


347 9-9*565 
82330 9.91556 


9 
9 
9 
8 


900 9.83822 


27 
27 
27 


0.16178 514 


26 
25 


35 


56760 9.75405 


68942 9.83849 


0.16151 1.4505 


36 


784 9.75423 


18 


314 9-9I547 


985 9-83876 


0.16124 496 


24 


37 


808 9.75441 


18 


297 9-9I538 


69028 9.83903 




0.16097 487 


2S 


38 


8 32 9-75459 




281 9.91530 




071 9-83930 


27 


0.16070 478 


22 


39 
40 


856 9-75478 


I 9 

18 
18 


264 9.91521 


9 
9 

8 


114 9-83957 


27 

27 
27 


0.16043 469 


21 


56880 9.75496 


82248 9.91512 


69157 983984 


0.16016 1.4460 


20 


41 


904 9-755I4 




231 9.91504 


9 
9 
9 
8 


200 9.8401 1 


0.15989 451 


19 


42 


928 9-75533 


I 9 

18 


214 991495 


243 9.84038 


27 
27 
27 
27 


0.15962 442 


18 


43 


952 9-75551 


18 


198 9.91486 


286 9.84065 


0.15935 433 


17 


44 


976 9-75569 


18 
18 


181 9.91477 


329 9.84092 


0.15908 424 


16 


45 


57000 9.75587 


82165 9.91469 


69372 9.841 19 


0.15881 1.4415 


15 


46 


024 9.75605 


19 
18 


148 9.91460 


9 
9 
9 
9 
8 


416 9.84146 


27 
27 
27 

27 
°6 


0.15854 406 


14 


47 


°47 9-75624 


132 9.91451 


459 9-84I73 


0.15827 397 


13 


48 


071 9-75642 


18 


115 9.91442 


502 9.84200 


O.15800 388 


12 


49 


°95 9-75660 


18 
18 
18 


098 9.91433 


545 9-84227 


0.15773 379 


11 

To 


50 


57i 19 9-75678 


82082 9.91425 


69588 9.84254 


0.15746 1.4370 


5i 


143 975696 


065 9.91416 


9 
9 
9 


631 9.84280 


27 

27 
27 

27 
27 


0.15720 361 


9 


52 


167 9.75714 




048 9.91407 


675 9-84307 


0.15693 352 


8 


53 


J 9i 9-75733 


I 9 
18 


032 9.91398 


718 9.84334 


0.15666 344 


7 


54 


215 9-75751 


18 
18 
18 


015 9-91389 


9 
8 


761 9.84361 


0.15639 335 


6 


55 


57238 9-75769 


81999 9.91381 


69804 9.84388 


0.15612 1.4326 


5 





262 9.75787 


982 9.91372 


9 


847 9-844I5 


0.15585 3i7 


4 


57 


286 9.75805 


965 9-91363 


9 


891 9.84442 


27 
27 
27 


0.15558 308 


3 


58 


310 9.75823 


18 


949 9-9I354 


9 
9 


934 9-84469 


0.15531 299 


2 


59 


334 9-7584I 


18 


932 9.91345 


977 9.84496 


0.15504 290 


1 


60 


353 9-75859 




915 9-9I336 


9 


70021 9.84523 


0.15477 281 







Nat. COS Log. d. 


Nat. Sin Log. 


d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


r 



55 c 











35 











f 


Nat. Sin Log. d. 


Nat. COS Log 


d. 


Nat.TanLog. c.d. Log. Cot Nat. 


60 





57358 975859 


18 


81915 9-9I336 


8 


70021 9.84523 i 0.15477 1.428 1 
064 9.84550 | £ 0.15450 273 


I 


381 9-75877 


18 


899 9-9 I 3 2 8 



9 
9 
9 
9 
9 

9 
8 


59 


2 


405 9-75895 


18 


882 9.91319 


107 9.84576 | „., 1 0.15424 264 


58 


3 


429 9.75913 


T8 


865 9.91310 


151 9.84603 


27 j 0.15397 255 

' 0.15370 246 

27 ! 0.15343 1-4237 
_ 0.15316 229 
' 0.15289 220 

2 6 O.15262 211 

26 


57 


4 


453 9-75931 


18 
18 


848 9.91301 


194 9.84630 


50 
55 


5 


57477 9-75949 


81832 9.91292 


70238 9.84657 


6 


501 9.75967 


18 


815 9.91283 


281 9.84684 


54 


7 


524 9.75985 


18 


798 9.91274 


325 9.8471 1 


53 


8 


548 9.76003 


18 


782 9.91266 


9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 


368 9.84738 


52 


9 


572 9.76021 


18 
18 


765 9.91257 


412 9.84764 


27 


0.15236 202 


5i 
50 


10 


57596 9-76039 


81748 9.91248 


70455 9.84791 


0.15209 1.4193 


II 


619 9.76057 


18 


731 9-91239 


499 9.84818 




0.15182 185 


49 


12 


643 9.76075 


18 


714 9.91230 


542 9.84845 




O.I5I55 J 7 6 


48 


13 


667 9.76093 


18 


698 9.91221 


586 9.84872 


27 


0.15128 167 


47 


14 


691 9.761 1 1 


18 

17 
18 


681 9.91212 


629 9.84899 


?5 0.15101 158 
nn - I 5°75 i.4 x 5° 


46 
45 


15 


57715 9.76129 


81664 9-9 I2 03 


70673 9.84925 


16 


738 9.76146 


647 9.91 194 


717 9.84952 ■ ~( 0.15048 141 

760 9.84979 ; 0.15021 132 


44 


17 


762 9.76164 


18 


631 9.91185 


43 


18 


786 9.76182 


iR 


614 9.91176 


804 9.85006 ' 


0.14994 124 


42 


19 


810 9.76200 


18 

iR 


597 9-91167 


848 9.85033 


26 


0.14967 115 


41 
40 


20 


57833 976218 


81580 9.91158 


70891 9.85059 


0.14941 1.4106 


21 


857 9-76236 


17 
18 


563 9.91 149 


935 9.85086 :' 


0.14914 097 


39 


22 


881 9.76253 


546 9.91141 


9 

9 
9 

% 

9 
9 
9 
9 
9 
9 
9 
10 

9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 
9 


979 9-85113 1 27 


0.14887 089 


38 


23 


904 9.76271 


18 


530 9.91132 


71023 9.85140 o £ 1 0.14860 080 


37- 


24 


928 9.76289 


18 

17 

tR 


513 9.91123 


066 9.85166 


" u 


0.14834 C7I 


36 
35 


25 


57952 976307 


81496 9.91114 


71110 9.85193 


27 


0.14807 1.4063 


26 


976 9.76324 


479 9-9II05 


154 9.85220 -* 
198 9.85247 | 27 


0.14780 054 


34 


27 


999 9-76342 


tR 


462 9.91096 


0.14753 045 


33 


28 


58023 9.76360 


18 


445 9-9I087 


242 9-85273 1 1 


0.14727 037 


32 


29 


047 9-76378 


17 

iR 


428 9.91078 


285 9-85300 


"1 

27 


0.14700 028 


31 
30 


30 


58070 9.76395 


81412 9.91069 


71329 9-85327 


0.14673 1.4019 


31 


094 9.76413 


tR 


395 9.91060 


373 9-85354 ; 26 


0.14646 on 


29 


32 


118 9.76431 


17 
18 


378 9.91051 


417 9.85380 f 


0.14620 002 


28 


33 


141 9.76448 


361 9.91042 


461 9.85407 ! ^7 


0.14593 1.3994 


27 


34 
35 


165 9.76466 
58189 9.76484 


18 

17 

iR 


344 9-91033 


505 9.85434 


26 


0.14566 985 


26 


81327 9.91023 


71549 9.85460 


0.14540 1.3976 


25 


36 


212 9.76501 


310 9.91014 


593 9-85487 i z 
637 9.85514 1 g 
681 9.85540 j 2° 


0.14513 968 


24 


37 


2 3 6 9765!9 


18 


293 9.91005 


0.14486 959 


23 


3« 


260 9.76537 


17 
18 

iR 


276 9.90996 


0.14460 951 


22 


39 
40 


283 976554 


259 9-90987 


725 9.85567 


27 


0.14433 942 


21 
20 


58307 9-76572 


81242 9.90978 


71769 9.85594 


0.14406 1.3934 


41 


33o 9-76590 


17 
iR 


225 9.90969 


813 9.85620 j ~ 


0.14380 925 


19 


42 


354 9.76607 


208 9.90960 


857 9.85647 S 


0.14353 916 


18 


43 


378 9.76625 


17 
18 

17 
18 


191 9.90951 


901 9.85674 


26 


0.14326 908 


17 


44 


401 9.76642 


174 9.90942 


946 9.85700 


27 


0.14300 899 


16 


45 


58425 9.76660 


81 157 9.90933 


71990 9.85727 


0.14273 1.3891 


15 


46 


449 9-76677 


140 9.90924 


72034 9.85754 


27 
26 


0.14246 882 


14 


47 


472 9.76695 




123 9.90915 


078 9.85780 




0.14220 874 


13 


48 


496 9.76712 


x 7 

18 


106 9.90906 


122 9.85807 


2 7 


0.14193 865 


12 


49 
50 


519 9.76730 


17 
18 


089 9.90896 


9 
9 
9 
9 
9 

9 


167 9.85834 


2 7 
26 


0.14166 857 


11 


58543 9-76747 


81072 9.90887 


72211 9.85860 


0.14140 1.3848 


10 


Si 


567 9-76765 


17 
18 


o55 9-90878 


255 9-85887 


27 
26 


0. 141 13 840 


9 


S2 


590 9.76782 


038 9.90869 


299 9-85913 




0.14087 831 


8 


53 


614 9.76800 




021 9.90860 


344 9.85940 


27 


0.14060 823 


7 


54 
55 


637 9.76817 


x 7 
18 


004 9.90851 


388 9.85967 


27 
26 


0.14033 814 


6 
~5~ 


58661 976835 


80987 9.90842 


72432 9.85993 


0.14007 1.3806 


56 


684 9.76852 


J 7 
18 


970 9.90832 


9 
9 

9 
9 


477 9.86020 


27 
26 


0.13980 798 


4 


57 


708 9.76870 




953 9-90823 


521 9.86046 




0.13954 789 


3 


f>8 


731 9.76887 




936 9.90814 


565 9.86073 


27 


0.13927 781 


2 


59 


755 9-76904 


I 7 
18 


919 9.90805 


610 9.86100 


27 
26 


0.13900 772 


1 


60 


779 9-76922 




902 9.90796 


654 9.86126 




0.13874 764 







Nat. COS Log. d. 


Nat. Sin Log. 


d. 


Nat. Cot Log. 


c.d.Log.TanNat. 


t 










54 


O 

















36 


O 








r 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







58779 9.76922 


17 
t8 


80902 9.90796 




72b54 9.86126 | „ 


0.13874 1.3764 


60 


i 


802 9.76939 


885 9.90787 


9 


699 9.86153 


26 


0.13847 755 


59 


2 


82b 9.76957 


17 
17 
18 

17 
17 
18 


8b 7 9.90777 




743 9-86179 




0.13821 747 


S8 


3 


849 9.76974 


850 9.90768 


9 


788 9.86206 


2 7 
2b 

27 

~b 


0.13794 739 


S7 


4 
5 


873 9-76991 


833 9-90759 


9 
9 
9 


832 9.86232 


0.13768 730 


56 
55 


5889b 9.77009 


80816 9.90750 


72877 9.86259 


0.13741 1.3722 





920 9.77026 


799 9-9074I 


921 9.86285 




0.13715 713 


54 


7 


943 9-77043 


782 9.90731 




966 9.86312 


27 
2b 


0.13688 705 


S3 


8 


967 9.77061 


17 
17 
17 

i 9 


765 9-90722 


9 


73010 9.86338 




0.13662 697 


S2 


9 


990 9.77078 


748 9.90713 


9 
9 


055 9-86365 


27 
27 

of, 


0.13635 688 


51 

50 


10 


59014 9.77095 


80730 9.90704 


73100 9.86392 


0.13608 1.3680 


ii 


°37 9-77 112 


713 9.90694 




144 9.86418 


2 ° 0.13582 b 7 2 


49 


12 


061 9.77130 


17 
17 
17 
18 


69b 9.90685 


9 
9 


189 9.86445 "a 1 0.13555 66 3 


48 


13 


084 9.77147 


679 9.90676 


234 9.86471 „„ 0.13529 655 


47 


I 4 


108 9.77164 


662 9.90667 


9 
10 


278 9.86498 


„g I 0.13502 647 
o „ ] O.I3476 I.3638 


46 
45 


15 


59131 9.77181 


8ob44 9.90657 


73323 9-86524 


ib 


154 9.77199 


17 
17 
17 
18 

17 
17 
17 
17 
17 
17 
17 
18 


b27 9.90648 


9 
9 
9 


368 9.86551 "' | 0.13449 630 


44 


17 


178 9.77216 


610 9.90639 


413 9.86577 6 | 0.13423 622 


43 


18 


201 9.77233 


593 9-90630 


457 986603 | 0.13397 613 


42 


19 


225 9.77250 


576 9.90620 


9 


502 9.86630 , J 
73547 9-86656 


0.13370 bos 


41 


20 


59248 9.7726.3 


80558 9.9061 1 


0.13344 1.3597 


40 


21 


272 9.77285 


541 9.90602 


9 


592 9-86683 1 % 


0.13317 588 


39 


22 


295 9-77302 


524 9-90592 




637 9.86709 




0.13291 580 


38 


23 


318 9-773I9 


507 9.90583 


9 


681 9.86736 


27 

2b 


0.13264 572 


37 


24 


342 9-77336 


489 9-90574 


9 
9 


726 9.86762 


27 

of 


0.13238 564 


3^ 


25 


59365 9-77353 


80472 9.90565 


73771 9.86789 


0.13211 1.3555 


35 


2b 


389 9-77370 


455 9-90555 




816 9.86815 


f O.13185 547 


34 


27 


412 9.77387 


438 9.90546 


9 
9 


861 9.86842 


2 6 0.13158 539 


33 


28 


436 9-77405 


17 
17 
17 
17 
17 


420 9.90537 


906 9.86868 


26 | O.13132 531 


32 


29 


459 9-77422 


403 9.90527 


9 


951 9.86894 


27 
26 


0.13106 522 


3i 
30 


30 


59482 9.77439 


80386 9.90518 


73996 9.86921 


0.13079 1.3514 


31 


50b 9.77456 


3b8 9.90509 


9 


74041 9.86947 




0-I3053 5o6 


29 


3 2 


529 9-77473 


351 9.90499 


9 


086 9.86974 


27 

of, 


0.13026 498 


28 


33 


552 9-77490 


334 9-90490 


131 9.87000 




0.13000 490 


27 


34 
35 


576 9-77507 


I 7 
17 


31b 9.90480 


9 


176 9.87027 


27 

2b 
"6 


0.12973 481 


2b 

25 


59599 9-77524 


80299 9.90471 


74221 9.87053 


0.12947 1.3473 


3t> 


622 9.77541 


17 
17 
17 
17 
17 
17 


282 9.90462 


9 


2b7 9.87079 




0.12921 465 


24 


37 


646 9-77558 


264 9.90452 




312 9.87106 


27 
26 


0.12894 457 


23 


3» 


669 9-77575 


247 9.90443 


9 


357 9-87I32 


^b 


0.12868 449 


22 


39 
40 


693 9-77592 


230 9.90434 


9 
10 

9 


402 9.87158 


27 

ob 


0.12842 440 


21 
20 


59716 9.77609 


80212 9.90424 


74447 9-87185 


0.12815 1.3432 


41 


739 9-77626 


195 9-904I5 


492 9.8721 1 




0.12789 424 


19 


42 


763 9.77643 


178 9.90405 





538 9.87238 


27 
2b 


0.12762 416 


18 


43 


78b 9.77660 


z 7 
17 
17 


ibo 9.90396 


9 


583 9.87264 


°b 


0.12736 408 


17 


44 


809 9.77677 


H3 9-90386 


9 


b28 9.87290 


27 
°b 


0.12710 400 


16 


45 


59832 9.77694 


80125 9.90377 


74674 9-873I7 


0.12683 1.3392 


15 


4 b 


85b 9.7771 1 


I 7 


108 9.90368 


9 


719 9.87343 


26 


0.12657 384 


14 


47 


879 9-77728 


J 7 

16 


091 9-90358 


10 


764 9-87369 


27 

26 


0.12631 375 


13 


48 


902 9.77744 


17 
17 
17 

17 
17 


073 9.90349 


9 


810 9.87396 


0.12604 367 


12 


49 


92b 9.77761 


056 9-90339 


9 


855 9-87422 


26 

27 
^6 


0.12578 359 


11 


50 


59949 9-77778 


80038 9.90330 


74900 9.87448 


0.12552 1.3351 


10 


5i 


972 9-77795 


021 9.90320 




946 9.87475 


0.12525 343 


9 


52 


995 9-778I2 


003 9.9031 1 


9 
10 


991 9.87501 


^b 


0.12499 335 


8 


53 


60019 9.77829 


79Q86 9.90301 




75037 9-87527 


27 
26 

06 


0.12473 327 


7 


54 


042 9.77846 


x 7 
16 


9b8 9.90292 


9 
10 


082 9.87554 


0.12446 319 


6 


55 


boob5 9.77862 


7995 1 9-90282 


75128 9.87580 


0.12420 1.33 1 1 


5 


56 


089 9.77879 


I 7 


934 9.90273 


9 


173 9.87606 


27 


0.12394 303 


4 


57 


112 9.77896 


I 7 


91b 9.90263 


10 


219 9.87633 


0.12367 295 


3 


5» 


135 9-779I3 


17 
16 


899 9.90254 


9 


264 9.87659 


-6 


0.12341 287 


2 


59 


158 9-77930 


881 9.90244 




310 9.87685 


~b 


0.12315 278 


1 


60 


182 9.77946 




864 9.90235 


9 


355 9-877ii 




0.12289 270 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


C.d. 


Log.TanNat. 


/ 



53 c 











37 


O 










f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. Log. Cot Nat. 







60182 9.77946 




79864 9.90235 




75355 9-877II 


27 
°6 


0.12289 


1.3270 


60 


I 


205 9-77963 


x 7 
17 
17 
16 


846 9.90225 


9 


401 9.87738 


0.12262 


262 


59 


2 


228 9.77980 


829 9.90216 


447 9-87764 


pf\ 


0.12236 


254 


58 


3 


251 9-77997 


811 9.90206 


9 
10 


492 9-87790 


27 
26 
06 


0.I22I0 


246 


57 


4 


274 9.78013 


17 


793 9-90I97 


538 9.87817 


O.I2I83 


238 


5b 
55 


5 


60298 9.78030 


79776 9.90187 


75584 9.87843 


O.I2I57 


1.3230 


6 


321 9.78047 


*7 
16 


758 9.90178 


9 


629 9.87869 


26 


0.I2I3I 


222 


54 


7 


344 9.78063 




741 9.90168 




675 9.87895 




O.I2I05 


214 


53 


8 


367 9.78080 


J 7 
17 
16 

17 


723 9.90159 


9 


721 9.87922 


^6 


0.12078 


206 


52 


9 
10 


390 9.78097 


706 9.90149 


10 
9 


767 9-87948 


26 
°6 


0.12052 


198 


5i 
50 


60414 9.781 13 


79688 9.90139 


75812 9.87974 


O.I2026 


1.3190 


ii 


437 9-78I30 


671 9.90130 


858 9.88000 




0.12000 


182 


49 


12 


460 9.78147 


I 7 
16 


653 9.90120 


9 


904 9.88027 


27 

°6 


O.II973 


175 


48 


13 


483 9.78163 




635 9.901 1 1 


950 9.88053 


26 


0.1 1947 


167 


47 


14 


506 9.78180 


J 7 
17 

16 


618 9.90101 


10 


996 9.88079 


26 
26 


0.11921 


159 


46 
45 


15 


60529 9.78197 


79600 9.90091 


76042 9.88105 


0.11895 


1-3151 


16 


553 978213 




583 9.90082 


9 


088 9.88131 




0.11869 


143 


44 


17 


576 9.78230 


x 7 
16 


565 9.90072 




134 9-88158 


27 
26 


0.11842 


135 


43 


18 


599 9.78246 




547 9-90063 


9 


180 9.88184 


26 


0.11816 


127 


42 


19 


622 9.78263 


I 7 
17 
16 


530 9.90053 


10 
9 


226 9.88210 


t 


0.1 1790 


119 


4i 
40 


20 


60645 9.78280 


79512 9.90043 


76272 9.83236 


0.1 1764 


1,3111 


21 


668 9.78296 




494 9-90034 


318 9.88262 1 ;r 


0.11738 


103 


39 


22 


691 9.78313 


x 7 

16 


477 9-90024 





364 9.88289 1 11 


0.11711 


095 


38 


23 


714 9.78329 




459 9-90014 




410 9.88315 ; ^ 


0.11685 


087 


37 


24 


738 9-78346 


I 7 

16 


441 9.90005 


9 

10 


456 9.88341 


26 


0.11659 


079 


36 


25 


60761 9.78362 


79424 9.89995 


76502 9.88367 


0.11633 


1.3072 


35 


26 


784 9.78379 


T 7 
16 


406 9.89985 




548 9.88393 1 ~ 


0.11607 


064 


34 


27 


807 9.78395 


388 9.89976 


9 


594 9.88420 J 


0.11580 


056 


33 


28 


830 9.78412 


J 7 
16 

17 
16 


371 9.89966 




640 9.88446 


26 


0.11554 


048 


32 


29 


853 9.78428 


353 9-89956 


9 


686 9.88472 


26 

°6 


0.11528 


040 


31 


30 


60876 9.78445 


79335 9-89947 


76733 9.88498 


0.11502 


1.3032 


30 


3i 


899 9.78461 




318 9.89937 




779 9.88524 


o£ 


0.1 1476 


024 


2Q 


32 


922 9.78478 


I 7 
16 
16 


300 9.89927 




825 9.88550 ~ 


0.1 1450 


017 


28 


33 


945 9-78494 


282 9.89918 


9 


871 9-88577 ! S 


0.11423 


009 


27 


34 


968 9.78510 


17 
16 


264 9.89908 


10 


918 9.88603 


26 


0.11397 


001 


26 

26 


35 


60991 9.78527 


79247 9-89898 


76964 9.88629 


0.11371 


1.2993 


36 


61015 9.78543 




229 9.89888 




77010 9.88655 ; ~ 


0.1 1345 


985 


24 


37 


038 9.78560 


z 7 
16 
16 

*7 

16 


211 9.89879 


9 


057 9.88681 


26 


0.11319 


977 


23 


38 


061 9.78576 


193 9.89869 




103 9.88707 


26 


0.11293 


970 


22 


39 
40 


084 9.78592 


176 9.89859 


10 


149 9.88733 


26 


0.1 1267 


962 


21 


61107 9.76009 


79158 9.89849 


77196 9-88759 


0.11241 


1.2954 


20 


4i 


130 9.78625 


140 9.89840 


9 


242 9.88786 


27 
26 


0.11214 


946 


19 


42 


153 9.78642 


I 7 
16 


122 9.89830 


10 


289 9.88812 


26 


0.11188 


938 


18 


43 


176 9.78658 


16 


105 9.89820 




335 9-88838 


26 


0.11162 


93i 


17 


44 
45 


199 9.7S674 




087 9.89810 


9 


382 9.88864 


26 
26 
26 


0.1 1 136 


923 


16 


"61222 9.76691 


J / 


79069 9.89801 


77428 9.88890 


O.IIIIO 


1-2915 


15 


4 6 


245 9-78707 


16 
16 


051 9.89791 




475 9.88916 


0.1 1084 


907 


14 


47 


268 9.78723 


033 9-8978I 


10 


521 9.8S942 


0.1 1058 


900 


13 


48 


291 9.78739 


016 9.89771 


10 


568 9.88968 z 


0.1 1032 


892 


12 


49 
50 


314 9.78756 


I 7 
16 
16 


78998 9.89761 


9 


615 9.88994 


26 


0.1 1 006 


884 


11 


61337 9.78772 


78980 9.89752 


77661 9.89020 


0.10980 


1.2876 


10 


Si 


360 9.78783 


962 9.89742 


10 


708 9.89046 1 ™ 


0.10954 


869 


9 


S2 


383 9.78805 


x 7 
16 
16 

16 

16 


944 9-89732 


10 


754 9.89073 11 
801 9.89099 | ^ 


0.10927 


861 


8 


.S3 


406 9.78821 


926 9.89722 




0.10901 


853 


7 


54 


429 9.78837 


908 9.89712 


10 


8 4 8 9.89125 ! 2b e 
77895 9.89151 1 


0.10875 


846 


6 


55 


61451 9.78853 


78891 9.89702 


0.10849 


1.2838 


5 


56 


474 9-78869 


873 9.89693 


9 


941 9-89177 ! 1 
9 s8 9.89203 ! f 


0.10823 


830 


4 


57 


497 9.78886 


li 

16 

16 


855 9.89683 


10 


0.10797 


822 


3 


.18 


520 9.78902 


837 9-89673 




78035 9.89229 ; 2° 


0.10771 


815 


2 


5Q 


543 9-789I8 


819 9.89663 


10 


082 9.89255 , f 
129 9.89281 | 2 ° 


0.10745 


807 


1 


BO 


566 9-78934 


801 9.89653 




0.10719 


799 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log.lc.d. Log.Ta 


n Nat. 


f 



52 c 









38 


O 








f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat. Tan Log. 


c.d. 


Log. Cot Nat. 







61566 9.78934 


t6 


78801 9.89653 




78129 9.89281 


06 


0.10719 1.2799 


60 


I 


589 9.78950 


17 
16 


783 9-89643 


10 


175 9-89307 


°6 


0.10693 792 


SQ 


2 


612 9.78967 


765 9.89633 




222 9.89333 


26 


0.10667 784 


S8 


3 


635 9.78983 


16 


747 9.89624 




269 9.89359 


°6 


0.10641 776 


S7 


4 


658 9.78999 


16 
16 


729 9.89614 


10 


316 9.89385 


26 
°6 


0.10615 769 


56 


5 


61681 9.79015 


787 1 1 9.89604 


78363 9.8941 I 


0.10589 1. 2761 


55 


6 


704 9.79031 


t6 


694 9.89594 


TO 


410 9.89437 


06 


0.10563 753 


54 


7 


726 9-79047 


t6 


676 9.89584 


IO 


457 9-89463 


°6 


0.10537 746 


53 


8 


749 9-79063 


16 


658 9-89574 


IO 


504 9.89489 


°6 


0.10511 738 


52 


9 
10 


772 9.79079 


16 
t6 


640 9.89564 


IO 


551 9-895*5 


26 

06 


0.10485 731 


5i 
50 


61795 9-79095 


78622 9.89554 


78598 9.89541 


0.10459 1.2723 


ii 


818 9.79111 


17 
16 


604 9.89544 




645 9.89567 


26 


0.10433 715 


49 


12 


841 9.79128 


586 9-89534 




692 9.89593 


og 


0.10407 708 


48 


13 


864 9.79144 


t6 


568 9.89524 




739 9.89619 


oft 


0.10381 700 


47 


14 


887 9.79160 


16 
t6 


550 9.89514 


IO 

9 


786 9.89645 


26 

26 


0.10355 693 


40 
45 


15 


61909 9.79176 


78532 9.89504 


78834 9.89671 


0.10329 1.2685 


ib 


932 9.79192 


16 


514 9-89495 


881 9.89697 


°6 


0.10303 677 


44 


17 


955 9-79208 


t6 


496 9-89485 




928 9.89723 


"6 


0.10277 670 


43 


18 


978 9.79224 


16 


478 9-89475 




975 9-89749 


<->6 


0.10251 662 


42 


19 


62001 9.79240 


16 

t6 


460 9.89465 


10 


79022 9.89775 


26 

26 


0.10225 6 55 


4i 
40 


20 


62024 9.79256 


78442 9-89455 


79070 9.89801 


0.10199 1.2647 


21 


046 9.79272 


16 


424 9.89445 




117 9.89827 


og 


0.10173 640 


39 


22 


069 9.79288 


16 


405 9.89435 




164 9.89853 


26 


0.10147 632 


38 


23 


092 9-79304 


15 
16 
16 


387 9.89425 




212 9.89879 


^6 


0.10121 624 


37 


2 4 


115 9-793I9 


369 9.89415 


IO 


259 9-89905 


26 
<^6 


0.10095 617 


30 


25 


62138 9-79335 


78351 9.89405 


79306 9.89931 


0.10069 1.2609 


35 


26 


160 9-79351 


16 


333 9-89395 




354 9-89957 


^6 


0.10043 602 


34 


27 


183 979367 


16 


315 9.89385 




401 9.89983 


16 


0.10017 594 


33 


28 


206 9.79383 


t6 


297 9-89375 




449 9.90009 


06 


0.09991 587 


32 


29 

30 


229 9-79399 


16 
16 


279 9-89364 


IO 


496 9.90035 


26 

25 
26 


0.09965 579 


3i 
30 


62251 9-794I5 


78261 9.89354 


79544 9-9006i 


0.09939 1-2572 


31 


274 9-79431 


16 


243 9-89344 


10 


591 9.90086 


0.09914 564 


29 


32 


297 9-79447 


16 


225 9-89334 


10 


639 9.90112 


<->6 


0.09888 557 


28 


33 


320 9.79463 


15 
16 
16 


206 9.89324 




686 9.90138 


26 


0.09862 549 


27 


34 
35 


342 9.79478 


188 9.89314 


IO 


734 9-90164 


26 

~6 


0.09836 542 


26 
25 


62365 9-79494 


78170 9.89304 


79781 9.90190 


0.09810 1.2534 


36 


388 9.79510 


16 


152 9.89294 


IO 


829 9.90216 


26 


0.09784 527 


24 


37 


411 9.79526 


16 


134 9.89284 


IO 


877 9.90242 


26 


0.09758 519 


23 


3« 


433 9-79542 


16 


116 9.89274 


IO 


924 9.90268 


06 


0.09732 512 


22 


39 
40 


456 9-79558 


15 
16 


098 9.89264 


IO 


972 9.90294 


26 

r>5 


0.09706 504 


21 


62479 9.79573 


78079 9.89254 


80020 9.90320 


0.09680 1.2497 


20 


41 


502 9.79589 


16 


061 9.89244 




067 9.90346 


25 
06 


0.09654 489 


19 


42 


524 9.79005 


16 


043 9.89233 




ii5 9-9037I 


0.09629 482 


18 


43 


547 9-7962I 


15 

16 
16 


025 9.89223 


TO 


163 9.90397 


^6 


0.09603 475 


17 


44 


570 9.79636 


007 9.89213 


IO 


211 9.90423 


26 
^6 


0.09577 467 


ib 
15 


45 


62592 9.79652 


77988 9.89203 


80258 9.90449 


0.09551 1.2460 


46 


615 9.79668 


16 


970 9.89193 


TO 


306 9.90475 


06 


0.09525 452 


14 


47 


638 9.79684 


15 
16 


952 9.89183 


IO 


354 9-9050I 


06 


0.09499 445 


13 


48 


660 9.79699 


934 9-89I73 




402 9.90527 


06 


0.09473 437 


12 


49 
50 


683 9-797I5 


16 

15 
16 


916 9.89162 


IO 


45o 9-90553 


25 
06 


0.09447 43° 


11 


62706 9.79731 


77897 9.89152 


80498 9.90578 


0.09422 1.2423 


10 


5i 


728 9.79746 


879 9.89142 


TO 


546 9.90604 


26 


0.09396 415 


9 


52 


751 9.79762 


16 


861 9.89132 


TO 


594 9.90630 


"6 


0.09370 408 


8 


53 


774 9-79778 


15 
16 
16 


843 9.89122 


TO 


642 9.90656 


06 


0.09344 401 


7 


54 


796 9.79793 


824 9.891 12 


II 


690 9.90682 


26 
26 


0.09318 393 


b 


55 


62819 9.79809 


77806 9.89101 


80738 9.90708 


0.09292 1.2386 


5 


5b 


842 9.79825 


15 
16 


788 9.89091 


TO 


786 9-90734 


25 
^6 


0.09266 378 


4 


57 


864 9.79840 


769 9.89081 


TO 


834 9-90759 


0.09241 371 


3 


sa 


887 9-79856 


16 


751 9.89071 




882 9.90785 


^6 


0.09215" 364 


2 


59 


909 9.79872 




733 9.89060 




930 9.9081 1 


^6 


0.09189 356 


1 


60 


932 9.79887 


I 5 


715 9-89050 




978 9.90837 




0.09163 349 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat. Cot Log. 


c.d. 


Log.TanNat. 


r 



51 









39° 








f 


Nat. Sin Log. d. 


Nat. COS Log. d. 


Nat.TanLog. 


c.d. 


Log. Cot Nat. 







62932 9.79887 


16 


77715 9.89050 j 




80978 9.90837 


06 


0.09163 1.2349 


60 


I 


955 9-79903 




696 9.89040 


10 


81027 9-90863 


^6 


0-09137 342 


59 


2 


977 9-79918 


16 


678 9.89030 




075 9.90889 


25 

06 


0.09111 334 


5* 


3 


63000 9.79934 


t6 


660 9.89020 




123 9.90914 


0.09086 327 


57 


4 


022 9.79950 


15 

16 


641 9.89009 


10 


171 9.90940 


26 

26 


0.09060 320 


50 
55 


5 


63045 9.79965 


77623 9.88999 


81220 9.90966 


0.09034 1.2312 


6 


068 9.79981 




605 9.88989 




268 9.90992 


05 


0.09008 305 


54 


7 


090 9.79996 


I 5 
16 


586 9.88978 




316 9.91018 




0.08982 298 


53 


8 


113 9.80012 


15 
16 


568 9.88968 




364 9.91043 


26 


0.08957 290 


52 


9 


135 9.80027 


550 9.88958 


10 


413 9.91069 


26 
06 


0.08931 283 


5i 
50 


10 


63158 9.80043 


77531 9.88948 


81461 9.91095 


0.08905 1.2276 


ii 


180 9.80058 


I 5 
16 


513 9.88937 




510 9.91121 


"6 


0.08879 268 


49 


12 


203 9.80074 


15 
16 


494 9.88927 




558 9.91147 




0.08853 261 


48 


13 


225 9.80089 


476 9.88917 




606 9.91 172 


25 
06 


0.08828 254 


47 


14 


248 9.80105 


15 

t6 


458 9.88906 


10 


655 9-91198 


26 
06 


0.08802 247 


40 
45 


15 


63271 9.80120 


77439 9-88896 


81703 9.91224 


0.08776 1.2239 


16 


293 9.80136 


15 

15 
16 


421 9.88886 




752 9.91250 


^6 


0.08750 232 


44 


17 


316 9.80151 


402 9.88875 




800 9.91276 


25 
~6 


0.08724 225 


4.3 


18 


338 9.80166 


384 9.88865 




849 9.91301 


0.08699 218 


42 


19 


361 9.80182 


15 
16 


366 9.88855 


11 


898 9.91327 


26 

-6 


0.08673 210 


4i 
40 


20 


633 8 3 9- 8oi 97 


77347 9-88844 


81946 9.91353 


0.08647 1.2203 


21 


406 9.80213 




329 9.88834 




995 9-9I379 


25 
^>6 


0.08621 196 


39 


22 


428 9.80228 


I 5 
16 


310 9.88824 




82044 9.91404 


0.08596 189 


33 


23 


451 9.80244 


15 
15 
16 


292 9.88813 




092 9.91430 


-6 


0.08570 181 


37 


2 4 


473 9.80259 


273 9.88803 


10 


141 9.91456 


26 

25 
~6 


0.08544 174 


30 
35 


25 


63496 9.80274 


77255 9-88793 


82190 9.91482 


0.08518 1.2167 


26 


518 9.80290 


15 


236 9.88782 




238 9.91507 


0.08493 l6 ° 


34 


27 


540 9.80305 


218 9.88772 




287 9-9I533 


26 


0.08467 153 


33 


28 


563 9.80320 


16 


199 9.88761 




336 9-9I559 


°6 


0.08441 145 


32 


29 


585 9.80336 


15 

15 

16 


181 9.88751 


10 


385 9-91585 


25 
"6 


0.08415 138 


31 


30 


63608 9.80351 


77162 9.88741 


82434 9.91610 


0.08390 1.2131 


30 


3? 


630 9.80366 


144 9.88730 




483 9.91636 


^6 


0.08364 124 


29 


32 


653 9.80382 


15 
15 

16 

15 
15 


125 9.88720 




531 9.91662 


05 


0.08338 117 


28 


33 


675 9-80397 


107 9.88709 




580 9.91688 


25 
26 


0.08312 109 


27 


34 


698 9.80412 


088 9.88699 


11 


629 9.91713 


0.08287 102 


26 
2T 


35 


63720 9.80428 


77070 9.88688 


82678 9.91739 


0.08261 1.2095 


36 


742 9.80443 


051 9.88678 




727 9.91765 ^ 


0.08235 088 


24 


37 


765 9.80458 


033 9.88668 




776 9.91791 




0.08209 081 


23 


3« 


787 9.80473 


J 5 
16 


014 9.88657 




82s 9.91816 


2 5 
26 


0.08184 074 


22 


39 


810 9.80489 


15 
15 
15 

16 
10 


76996 9.88647 


11 


874 9.91842 


26 
25 


0.08158 066 


21 


40 


63832 9.80504 


76977 9.88636 


82923 9.91868 


0.08132 1.2059 


20 


41 


854 9.80519 


959 9.88626 




972 9.91893 


0.08107 052 


19 


42 


877 9.80534 


940 9.88615 




83022 9.91919 I 26 


0.08081 045 


18 


43 


899 9.80550 




921 9.88605 




07I 9.91945 oA 


0.08055 038 


17 


44 
45 


922 9.80565 


15 
15 


903 9.88594 


10 


120 9.91971 


25 
06 


0.08029 031 


16 


63944 9.80580 


76884 9.88584 


83169 9.91996 


0.08004 1.2024 


15 


46 


966 9.80595 


866 9.88573 




218 9.92022 


06 


0.07978 017 


14 


47 


989 9.80610 


I 5 

15 
16 


847 9.88563 




268 9.92048 


25 

26 


0.07952 009 


13 


48 


640 1 1 9.80625 


828 9.88552 




317 9.92073 


0.07927 002 


12 


49 


033 9.80641 


15 


810 9.88542 


11 


366 9.92099 


26 

25 
26 


0.07901 1. 1995 


11 


50 


64056 9.80656 


76791 9.88531 


83415 9.92125 


0.07875 1.1988 


10 


51 


078 9.80671 


I 5 


772 9.88521 




465 9.92150 


0.07850 981 


9 


52 


100 9.80686 


x 5 


754 9-88510 




514 9.92176 


26 


0.07824 974 


8 


53 


123 9.80701 


I 5 
15 
15 


735 9-88499 




564 9.92202 




0.07798 967 


7 


54 
55 


145 9.80716 


717 9.88489 


11 


613 9.92227 


26 
"6 


0-07773 960 


6 
5 


64167 9.80731 


76698 9.88478 


83662 9.92253 


0.07747 1.1953 


5^ 


190 9.80746 ;g 


679 9.88468 




712 9.92279 




0.07721 946 


4 


57 


212 9.80762 *° 


661 9.88457 




761 9.92304 


' ^6 


0.07696 939 


3 


5« 


234 9.80777 


x 3 


642 9.88447 




811 9.92330 


1 26 


0.07670 932 


2 


59 


256 9.80792 


I 5 


623 9.88436 


" 1 860 9-92356 




0.07644 925 


1 


60 


279 9.80807 


z 5 


604 9.88425 


| 910 9.92381 


2 5 


0.07619 918 







Nat. COS Log. d. 


Nat. Sin Log. d. 


Nat.CotLogJcd 


Log.TanNat 


t 



50 



4jr 

Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog. c.d 



Log. Cot Nat 



30 

31 
32 
33 
34 
35 
36 
37 
33 
39 
40 

4i 
42 

43 
44 
45 
46 

47 
48 

49 
50 
5* 
52 
53 
54 
55 
56 
57 
53 
59 
60 



64279 
301 
323 

346 
368 



9.80807 
9.80822 
9.80837 
9.80852 
9.80867 



64390 
412 

435 
457 
479 



9.80882 
9.80897 
9.80912 
9.80927 
9.80942 



64501 
524 
546 
568 
590 



9.80957 
9.80972 
9.80987 
9.81002 
9.81017 



64612 
635 
657 
679 
701 



9.81032 
9.81047 
9.81061 
9.81076 
9.81091 



64723 
746 
768 
790 
812 



9.81 106 
9.81121 
9.81136 
9.81151 
9.81 166 



64834 
856 
878 
901 
923 



9.81180 

9-8II95 
9.81210 
9.81225 
9.81240 



64945 
967 
989 

6501 1 
033 



9.81254 
9.81269 
9.81284 
9.81299 
9.81314 



65055 
077 
100 
122 
144 



9.81328 

9-81343 
9.81358 
9.81372 
9.81387 



65166 
188 
210 
232 
254 



9.81402 
9.81417 

9-8I43I 
9.81446 
9.81461 



65276 
298 
320 
342 
364 



9.81475 
9.81490 
9.81505 
9-8I5I9 
9-81534 



65386 
408 
430 
452 
474 



9.81549 
9.81563 
9.81578 
9.81592 
9.81607 



65496 
5i8 
540 
562 

584 
606 



9.81622 
9.81636 
9.81651 
9.81665 
9.81680 
9.81694 J 



76604 
586 
567 
548 
530 



9.88425 
9.88415 
9.88404 
9.88394 
9-88383 



765 1 1 
492 
473 
455 
436 



9.88372 
9.88362 

9.88351 
9.88340 
9.88330 



76417 
398 
380 
361 
342 

76323 
304 
286 
267 
248 



9.88319 
9.88308 
9.88298 
9.88287 
9.88276 
9.88266 
9.88255 
9.88244 
9.88234 
9.88223 



76229 
210 
192 
173 
154 



9.88212 
9.88201 
9.88191 
9.88180 
9.88169 



76135 
116 
097 
078 
059 



9.88158 
9.88148 
9.88137 
9.88126 
9.88115 



76041 
022 
003 

75984 
965 



9.88105 
9.88094 
9.88083 
9.88072 
9.88061 



75946 
927 
908 
889 
870 



9.88051 
9.88040 
9.88029 
9.88018 
9.88007 



75851 
832 
813 

794 
775 



9.87996 
9.87985 

987975 
9.87964 

9-87953 



75756 
738 
719 
700 
680 



9.87942 
9.87931 
9.87920 
9.87909 
9.87898 



75661 
642 
623 
604 
585 



9.87887 
9.87877 
9.87866 
9-87855 
9-87844 



75566 
547 
528 
509 
490 
471 



9.87833 
9.87822 
9.87811 
9.87800 
9.87789 
9.87778 



83910 

960 

84009 

059 
108 



9.92381 
9.92407 

992433 
9.92458 
9.92484 



84158 
208 
258 
3°7 
357 



9.92510 

9.92535 
9.92561 
9.92587 
9.92612 



84407 
457 
507 
556 
606 



9.92638 
9.92663 
9.92689 
9.92715 
9.92740 



84656 
706 
756 
806 
856 



9.92766 
9.92792 
9.92817 
9.92843 
9.92868 



84906 

956 

85006 

o57 
107 



9.92894 
9.92920 
9.92945 
9.92971 
9.92996 



85157 
207 

257 
308 
358 



9.93022 
9.93048 

9-93073 
9.93099 
9.93124 



85408 
458 
509 
559 
609 



9-93I50 
9-93175 
9.93201 
9.93227 
993252 



85660 
710 
761 
811 
862 



9.93278 
9.93303 
9.93329 
9-93354 
9.93380 



85912 

963 

86014 

064 

"5 



9.93406 
9-93431 
9-93457 
9.93482 
9-93508 



86166 
216 
267 
318 
368 



9-93533 
9-93559 
993584 
9.93610 
9.93636 



86419 
470 
521 
572 
623 



9.93661 
9.93687 
9.93712 
9-93738 
993763 



86674 
725 
776 
827 



993789 
9.93814 
9.93840 
993865 
9.93891 
9.93916 



0.07619 

0.07593 
0.07567 
0.07542 
0.07516 



1.1918 
910 
903 



0.07490 
0.07465 
0.07439 
0.07413 
0.07388 



1.1882 

875 
868 
861 
854 



0.07362 

0.07337 
O.0731 1 
0.07285 
0.07260 



1. 1847 
840 

833 
826 
819 



0.07234 
0.07208 
0.07183 
O.07157 
0.07132 



1.1812 
806 
799 
792 
785 



O.07106 
0.07080 
0.07055 
O.07029 
0.07004 



1.1778 
771 
764 

757 
75° 



0.06978 
0.06952 
0.06927 
O.06901 
O.06876 



I-I743 
736 
729 
722 

715 



O.06850 
0.06825 
O.06799 
0.06773 
0.06748 



1. 1708 
702 

695 
688 
681 



O.06722 
O.06697 
O.06671 
O.06646 
O.06620 



1. 1674 
667 
660 
653 
647 



O.06594 
O.06569 
O.06543 
O.06518 
O.06492 



1. 1640 

633 
626 
619 
612 



O.06467 
O.06441 
O.06416 
O.06390 
O.06364 



1.1606 
599 
592 
585 
578 



0.06339 
0.06313 
0.06288 
0.06262 
0.06237 



I.I57I 

565 
558 
551 
544 



0.0621 1 
0.06186 
0.06160 
0.06135 
0.06109 
0.06084 



I-I538 
53i 
524 
517 
5io 
5°4 



Nat. COS Log. d. Nat. Sin Log. d. Nat. Cot Log. c.d. Log.TanNat. / 

49° 



41^ 

' Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog. c.d. Log.Cot Nat 



65606 
628 
650 
672 
694 

65716 
738 
759 
781 
803 



^25 



913 



9.81694 j 
9-8I709 2 

9.81738 j \l 
9-8I752 ! W 
9.81767 I * 

9-81781 i \\ 
9-81796 1 2 

9.81810 Jz 

9-81825 



*•— q ! I4 

„ 9.81839 !- 

847 9.81854 ! 2 
869 9.81868 : r[ 

9.81882 1 J4 

9.81897 ! ^ 
9.81911 ; 3 
9.81926 g 
9.81940 ' i 

9.81955 Jj 
9.81969 + 

9-81983 \\ 
9.81998 2 

9.82012 ; _J 

9.82026 ; I* 



65935 
956 
978 

66000 
022 



66044 

066 

088 

109 

131 



66153 

175 
197 
218 
240 



9.82041 



9-82055 ' JJ 
9.82069 ^3 
9.82084 2 
9.82098 L + 
9.82112 i4 

-^ T 1 



66262 
284 
306 
3^ 

349 



66371 

393 

414 

436 

458 



9.82126 T T 

9.82141 i 2 

9-82155 I 2 
9.82169 JJ 
9-82184 ^ 



66480 
501 
523 
545 
566 



9.82198 
9.82212 
9.82226 
9.82240 
9-82255 



66588 
610 
632 
653 

675 



66697 
718 
740 
762 
783 



66805 
827 
848 
870 
891 
913 



y— jj 14 
9.82269 J 
9-82283 g 

9.82297 2 
9-82311 ' jj 
9-82326 ^ 



y T^ , 14 

9.82340 

9-82354 2 
9.82368 
9.82382 ; \\ 

9-82396 ! H 

-= 1 14 



- y T oy " : i 4 
9.82410 J 
9.82424 I + 
9.82439 2 
9-82453 Z 
9-82467 *;+ 
9.82481 j 

9-82495 il 
9.82509 + 
9.82523 11 

9.82537 2 

9.8255I I4 



75471 9.87778 

452 9.87767 

433 9-87756 

414 9-87745 

395 9-87734 



75375 9-87723 

356 9.87712 

337 9-8770I 

318 9.87690 

299 9.87679 



75280 9.87668 

261 9.87657 

241 9.87646 

222 9.87635 

203 9.87624 



75184 9.87613 

165 9.87601 

146 9.87590 

126 9.87579 I 

107 9.87568 



75088 9-87557 

069 9.87546 

050 9-87535 

030 9-87524 

on 9.87513 



74992 9.87501 

973 9-87490 

953 9-87479 

934 9.87468 

915 9-87457 



74896 9.87446 

876 9.87434 

857 9.87423 

838 9.87412 

~ 818 9.87401 



74799 9.87390 

780 9.87378 

760 9.87367 

741 9.87356 

722 9.87345 



74703 9.87334 

683 9.87322 

664 9.8731 1 

644 9.87300 

625 9.87288 



74606 9.87277 

586 9.87266 

567 9-87255 

548 9.87243 

528 9.87232 



74509 9.87221 ; 

489 9.87209 

470 9.87198 

451 9.87187 

431 9-87175 



74412 9.87164 

392 9-87153 

373 9-87141 

353 9-87I30 

334 9-87119 

314 9.87107 



86929 
980 

87031 
082 
133 



9.93916 
9-93942 
9-93967 
9-93993 
9.94018 



87184 
236 
287 
338 
389 



9.94044 
9.94069 
9.94095 
9.94120 
9.94146 



87441 
492 
543 
595 
646 



9.94171 
9.94197 
9.94222 
9.94248 
9-94273 



87698 

749 
801 
852 
904 



9.94299 
9-94324 
9-94350 
9-94375 
9.94401 



87955 
88007 

o59 
no 
162 



9.94426 
9-94452 
9.94477 

9-94503 
9.94528 



88214 
265 
317 
369 
421 



9-94554 
9-94579 
9.94604 
9.94630 
9-94655 



0.06084 i^SCH 

0.06058 497 

0.06033 49o 

0.06007 483 

0.05982 477 



0.05956 1. 1470 
0.05931 463 
0.05905 456 
0.05880 450 
0.05854 443 



0.05829 1.1436 
0.05803 430 

! 0.05778 423 

i 0.05752 410 

0.05727 410 



0.05701 1. 1403 
0.05676 396 
0.05650 389 
0.05625 383 
0-05599 3 76 
0.05574 1. 1369 
0.05548 363 
0.05523 356 

0.05497 349 
0.05472 343 



88473 
524 
576 
628 
680 



9.94681 
9.94706 

9-94732 
9-94757 
9-94783 



0.05446 1.1336 
0.05421 329 
0.05396 323 
0.05370 316 
11 1 0-05345 3io 



I 0.05319 1. 1303 

0.05294 296 

0.05268 290 

0.05243 283 

0.05217 276 



88732 
784 
836 
888 
940 



9.94808 
9.94834 
9.94859 
9.94884 
9.94910 



88992 

89045 
097 
149 
201 



9-94935 
9.94961 
9.94986 
9.95012 
9-95037 



89253 
306 
358 
410 

463 



9.95062 
9.95088 
9-951 13 
9-95I39 
9-95i64 



567 
620 
672 

725 



9.95190 

9-952I5 
9.95240 
9.95266 
9.95291 



89777 
830 
883 

935 

988 

90040 



9-95317 
9-95342 
995368 

9-95393 
9.95418 

9-95444 



26 



26 



0.05192 1. 1270 
0.05166 263 
0.05141 257 
0.05116 250 
0.05090 243 



0.05065 1.1237 
0.05039 230 
0.05014 224 
0.04988 217 
0,04963 211 



0.04938 1.1204 
0.04912 197 
0.04887 191 
0.04861 184 
0.04836 178 



0.04810 1.1171 
0.04785 165 
0.04760 158 
0.04734 152 
0.04709 



145 



0.04683 1. 1 139 
0.04658 132 
0.04632 126 
0.04607 119 
0.04582 113 
0.04556 106 



Nat. COS Log. d. 



Nat. Sin Log. d. 

4tf 



Nat. Cot Log.cd. Log.TanNat 



42; 

Nat. Sin Log. d. Nat. COS Log. d. Nat.TanLog.lc.d 



Log. Cot Nat. 



66913 
935 
956 
978 
999 



9.82551 
9.82565 
9.82579 
9-82593 
9.82607 



67021 

043 
064 
086 
107 



9.82621 
9.82635 
9.82649 
9.82663 
9.82677 



67129 

151 
172 
194 
215 



9.82691 
9.82705 
9.82719 

9-82733 
9.82747 



67237 
258 
280 
301 
3 2 3 



9.82761 
9.82775 
9.82788 
9.82802 
9.82816 



67344 
366 

387 
409 

430 



9.82830 
9.82844 
9.82858 
9.82872 
9.82885 



67452 
473 
495 
5i6 
538 



9.82899 
9.82913 
9.82927 
9.82941 
9-82955 



67559 
580 
602 
623 
645 



9.82968 
9.82982 
9.82996 
9.83010 
9.83023 



67666 
688 
709 
730 
75 2 



9-83037 
9.83051 
9.83065 
9-83078 
9.83092 



67773 
795 
816 

837 
859 



9.83106 
9.83120 

9-83I33 
9.83147 
9.83161 



7880 
901 
923 
944 
965 



9.83174 
9.83188 
9.83202 
9.83215 
9-83229 



67987 

68008 

029 

051 

072 



9.83242 
9.83256 
9.83270 
9.83283 
9-83297 



68093 

115 
136 

157 
179 
200 



9.83310 
9-83324 
9-83338 
9-8335J 
9-83365 
9-83378 



743 14 
295 
276 
256 
237 



9.87107 
9.87096 
9.87085 
9.87073 
9.87062 



74217 
198 
178 
159 
139 



9.87050 
9.87039 
9.87028 
9.87016 
9.87005 



74120 
100 
080 
061 
041 



9.86993 
9.86982 
9.86970 
9.86959 
9.86947 



74022 
002 

73983 
963 
944 

73924 
904' 
885 
865 
846 

73826 
806 
787 
767 
747 



9.86936 
9.86924 
9.86913 
9.86902 
9.86890 
9.86879 
9.86867 
9.86855 
9.86844 
9.86832 
9.86821 
9.86809 
9.86798 
9.86786 
9.86775 



73728 
708 



649 



9.86763 
9.86752 
9.86740 
9.86728 
9.86717 



73629 
610 
59o 
570 
551 



9.86705 
9.86694 
9.86682 
9.86670 
9-86659 



73531 
5" 
491 
472 
452 



9.86647 
986635 
9.86624 
9.86612 
9.86600 



73432 
4i3 
393 
373 
353 



9.86589 
9.86577 
9.86565 

9-86554 
9.86542 



73333 
314 
294 

274 
254 



9.86530 
9.86518 
9.86507 
9.86495 
9.86483 



73234 
215 
195 
175 
155 
135 



9.86472 
9.86460 
9.86448 
9.86436 
9.86425 
9.86413 



90040 

093 

146 
199 
251 



9-95444 
9.95469 

9-95495 
9-9552? 
9-95545 



90304 

357 
410 

463 
5i6 



9-95571 
9-95596 
9.95622 
9.95647 
9.95672 



90569 
621 
674 
727 
781 

90834 
887 
940 

993 
91046 



9.95698 
9-95723 
9-95748 
9-95774 
9-95799 
9-95825 
9-95850 
9-95875 
9.95901 
9.95926 



91099 

i53 
206 

259 
3i3 



9-95952 
9-95977 
9.96002 
9.96028 
9-96053 



91366 
419 

473 
526 
580 



9.96078 
9.96104 
9.96129 

9-96I55 
9.96180 



91633 
687 
740 
794 
847 



9.96205 
9.96231 
9.96256 
9.96281 
9.96307 



91901 
955 



062 
116 



9.96332 
9-96357 
996383 
9.96408 

9-96433 



92170 
224 
277 
33i 
385 



996459 
9.96484 
9.96510 

996535 
9.96560 



92439 
493 
547 
601 

655 

92709 
763 
817 
872 
926 



9.96586 
9.9661 1 
9.96636 
9.96662 
9.96687 



9.96712 
9.96738 
9.96763 
9.96788 
9.96814 



92950 

93034 
088 

143 
197 
252 



9.96839 
9.96864 
9.96890 
9.96915 
9.96940 
9.96966 



0.04556 
O.04531 
0.04505 
O.04480 
0.04455 



1.1106 
100 

093 
087 
080 



0.04429 
0.04404 
O.04378 
004353 
0.04328 



1. 1074 
067 
061 

054 
048 



0.04302 
0.04277 
0.04252 
0.04226 
0.04201 



1.1041 

035 
028 
022 
016 



0.04175 
O.04150 
0.04125 
0.04099 
0.04074 



1. 1009 
003 

1.0996 
990 
983 



0.04048 
0.04023 
0.03998 
0.03972 
0.03947 



1.0977 
971 
964 
958 
951 



0.03922 
O.03896 
0.03871 
0.03845 
0.03820 



1.0945 
939 
93 2 
926 
919 



0.03795 
O.03769 
0.03744 
0.03719 
0.03693 



.0913 
907 
900 



0.03668 
0.03643 
0.03617 
0.03592 
0.03567 



1.0881 

875 
869 
862 
856 



0.03541 
0.03516 
0.03490 
0.03465 
0.03440 



1.0850 
843 
837 
831 
824 



0.03414 
0.03389 
0.03364 
0.03338 
003313 



.0818 
812 
805 
799 
793 



0.03288 
0.03262 
0.03237 
0.03212 
0.03186 



1.0786 
780 

774 
768 
761 



0.03161 
0.03136 
0.03110 
0.03085 
0.03060 
0.03034 



I-0755 
749 
742 

736 
730 
724 



Nat. COS Log. d. 



Nat. Sin Log. d. 

47 



Nat. Cot Log. c.d 



Log.TanNat. ' 



43 c 

Nat. Sin Log. d. Nat. COS Log. d. 



Nat. Tan Log. 



c.d 



Log. Cot Nat 



40 
41 
42 

43 
44 



68200 
221 
242 
264 
285 



68306 
327 
349 
37° 
39i 



9-83378 
9.83392 

9-83405 
9.83419 
9-83432 



68412 
434 
455 
476 

497 



14 
13 
14 
13 

9-83446 I \\ 

9 83459 l 2 
9-83473 \\ 
9.83486 \\ 

9-85500 T t 



68518 
539 
56i 
582 
603 



9-835I3 x l 

9-83527 It 

9-83540 \l 

9.83554 \\ 

9-83567 \ 3 



9.835^1 



645 
666 
688 
709 



68730 
75i 
772 

793 
814 



14 

9-83594 \\ 
9.83608 \\ 
9-8362I I 11 
9-83634 ** 
9-83648 £ 
9.83661 J3 

9-83674 g 
9-83688 \\ 

9-8370I 1 3 



68835 
857 



920 



9.83715 £ 
9-83728 I 11 
9-83741 I \l 
9-83755 ' Jt 
9-8376S I 3 



68941 
962 

983 

69004 

025 



69046 
067 
088 
109 
130 



9.83781 

9-83795 ^ 
9-838od t 3 
9-83821 I g 
9-83834 ! I 3 
9-8384S I * 
9.83861 I 3 
9.83874 g 
9.838S 7 I \\ 
9.83901 ] 



69151 
172 

193 
214 

235 



9.839I4 * 

9.83927 3 

9.83940 T *J 

9-83954 ^ 

9.83967 2 



69256 
277 
298 
319 
340 



69361 
382 
403 
424 
445 
466 



9.83980 I 

9-83993 3 

9.84006 I 3 

9.84020 J 

9.84033 ^ 

9.84046 ^ 

9-84059 ^ 

9.84072 ^ 

9-84085 *f 

9-84098 £ 

9.84112 I 

9-84125 g 

9.84138 11 

9.84I5I J| 

9.84164 J3 

9.84177 ^ 



73135 
116 
096 
076 
056 



9.86413 
9.86401 
9.86389 

9-86377 
9.86366 



73036 
016 

72996 
976 
957 



9-86354 
9.86342 
9.86330 
9.86318 
9.86306 



72937 
917 
897 
877 
857 



9.86295 
9.86283 
9.86271 
9.86259 
9.86247 



72837 
817 
797 
777 

757 



9.86235 
9.86223 
9.86211 
9.86200 
9.86188 



72737 
717 
697 
677 
657 



9.86176 
9.86164 
9.86152 
9.86140 
9.86128 



72637 
617 
597 
577 
557 



9.861 16 
9.86104 
9.86092 
9.86080 
9.86068 



72537 
5i7 
497 
477 
457 



9.86056 
9.86044 
9.86032 
9.86020 
9.86008 



72437 
417 

397 
377 
357 
72337 
317 
297 
277 
257 



9.85996 
9.85984 
9.85972 
9.85960 
9.85948 

9.85936 
9.85924 
985912 
9.85900 
9.85888 



72236 
216 
196 
176 
1^6 



9.85876 
9.85864 
9.85851 
9-85839 
9-85827 



72136 
116 

095 
075 

°55 



9.85815 
9.85803 
9.85791 

9-85779 
9.85766 



72035 
015 

71995 
974 
954 
934 



9-85754 
9.85742 

9.85730 
9.85718 
9.85706 
9-85693 



93252 
306 
360 

415 
469 



9.96966 
9.96991 
9.97016 
9.97042 
9.97067 



93524 
578 
633 
688 

742 



9.97092 
9.971 18 

9-97I43 
9.97168 

9-97*93 



93797 
852 
906 
961 

94016 



9.97219 
9.97244 
9.97269 

997295 
9.97320 



94071 
125 
180 

235 
290 



9-97345 
9-97371 
9-97396 
9.97421 

9-97447 



94345 
400 

455 
5io 
565 



9.97472 
9.97497 
9-97523 
9-97548 
9-97573 



94620 
676 

73i 
786 
841 



9-97598 
9.97624 
9.97649 
9.97674 
9.97700 



94896 
952 

95007 
062 
118 



9.97725 
9-97750 
9.97776 
9.97801 
9.97826 



95173 
229 
284 
340 
395 



9.97851 
9.97877 
9.97902 
9.97927 
9-97953 



9545i 
506 
562 
618 
673 



9.97978 
9.98003 
9.98029 
9.98054 
9.98079 



95729 
785 
841 
897 
952 



9.98104 
9.98130 

9-98I55 
9.98180 
9.98206 



004 
120 
176 
232 



9.98231 
9-98256 j 2 
9.98281 H 
9-98307 2 , 
9-98332 5 



344 
400 

457 
5i3 
569 



9-98357 
9-98383 
9.98408 

998433 
9.98458 
9.98484 



0.03034 
0.03009 
0.02984 
0.02958 
Q-02933 
0.02908 
0.02882 
0.02857 
0.02832 
0.02807 



1.0724 
717 
711 
705 
699 

1.0692 
686 
680 
674 



0.02781 
0.02756 
0.02731 
0.02705 
0.02680 



1. 066 1 

655 
649 

643 
637 



0.02655 
0.02629 
0.02604 
0.02579 
0.02553 



1.0630 
624 
618 
612 
606 



0.02528 
0.02503 
0.02477 
0.02452 
0.02427 



1.0599 
593 
587 
581 

575 



0.02402 
0.02376 
0.02351 
0.02326 
0.02300 



1.0569 
562 
556 
55° 
544 



0.02275 
0.02250 
0.02224 
0.02199 
0.02174 



1.0538 
532 
526 
5i9 
513 



0.02149 
0.02123 
0.02098 
0.02073 
0.02047 



1.0507 
501 
495 
489 

483 



0.02022 
0.01997 
0.01971 
0.01946 
0.01921 



1.0477 
470 
464 
458 
452 



0.01896 
0.01870 
0.01845 
0.01820 
0.01794 



1.0446 
440 

434 
428 
422 



0.01769 
c.01744 
0.01719 
0.01693 
0.01668 



1.0416 
410 
404 
398 
392 



0.01643 
O.01617 
0.01592 
0.01567 
0.01542 
0.01516 



1.0385 
379 
373 
367 
361 

355 



Nat. COS Log. d.|Nat. Sin Log. d. 

4fr 



Nat. Cot Log. I c.d. Log.Tan Nat 



' Nat. Sin Log. d. 



44= 

Nat. COS Log. d. Nat. Tan Log. 



.d. Log. Cot Nat 



487 
508 
529 
549 



9.84177 
9.84190 
9.84203 
9.84216 
9.84229 



69570 

59i 
612 

633 
654 



9.84242 

984255 
9.84269 
9.84282 
9.84295 



69675 
696 
717 
737 
758 



9.84308 
9-8432I 
9-84334 
9.84347 
9.84360 



69779 
800 
821 
842 
862 



9-84373 
9-84385 
9-84398 
9.8441 1 
9.84424 



904 

925 
946 
966 



9-84437 
9.84450 
9.84463 
9.84476 
9.84489 



69987 

70008 

029 

049 

070 



9.84502 

9-845I5 
9.84528 
9.84540 
9-84553 



70091 
112 
132 

153 
174 



9.84566 

9-84579 
9.84592 
9.84605 
9.84618 



70I9S 
215 
236 
257 
277 



9-84630 
9.84643 
9.84656 
9.84669 
9.84682 



70298 
3i9 
339 
360 
381 



9.84694 
9.84707 
9.84720 
9-84733 
9-84745 



70401 
422 
443 
463 
484 



9.84758 
9.84771 

9.84784 
9.84796 
9.84809 



70505 
525 
546 
567 
587 



9.84822 

9-84835 
9.84847 
9.84860 
9.84873 



70608 
628 
649 
670 
690 
711 



9.84885 
9.84898 
9.8491 1 
9.84923 
9.84936 
9.84949 



71934 
914 
894 
873 
853 



9-85693 
9.85681 
9.85669 
9.85657 
9-85645 



71833 
813 
792 
772 
752 



9.85632 
9.85620 
9.85608 
9-85596 
9-85583 



71732 
711 
691 
671 
650 



9-85571 
9-85559 
9-85547 
9-85534 
9.85522 



71630 
610 
59o 
569 
549 



9.85510 
9-85497 
9-85485 
9-85473 
9.85460 



71529 
508* 
488 
468 
447 



9.85448 
9-85436 
9-85423 
9.8541 1 

9-85399 



71427 

407 
386 
366 
345 



9.85386 

9-85374 
9.85361 

9-85349 
9-85337 



71325 
305 
284 
264 
243 



9-85324 
9.85312 
9.85299 
9.85287 

9.85274 



71223 
203 
182 
162 
141 



9.85262 
9.85250 

9-85237 
9.85225 
9.85212 



71121 
100 
080 
059 
039 



9.85200 
9.85187 

985175 
9.85162 
9.85150 



71019 



978 
957 
937 



9-85I37 
9.85125 
9.851 12 
9.85100 
9.85087 



70916 
896 
875 
855 
834 



9.85074 
9.85062 
9.85049 

9.85037 
9.85024 



70813 
793 
772 
752 
73 1 
711 



9.85012 
9.84999 
9.84986 
9.84974 
9.84961 
9.84949 



96569 
625 
681 
738 
794 



9.98484 
9.98509 

9-98534 
9.98560 

9-98585 



96850 
907 
9°3 

97020 
076 



9.98610 

9.98635 
9.98661 
9.98686 
9.98711 



97133 
189 
246 
302 
_359 
97416 
472 

529 
586 

643 



998737 
9.98762 
9.98787 
9.98812 
^.98832 
9.98863 
9.98888 
9.98913 

9-98939 
9.98964 



97700 
756 
813 
870 
927 



9.98989 
9.99015 
9.99040 
9.99065 
9.99090 



97984 

98041 

098 

155 
213 



9.991 16 
9.99141 
9.99166 
9.99191 
9.99217 



98270 
327 
384 
441 

499 



9.99242 
9.99267 
9.99293 
9.99318 
9-99343 



98556 
613 
671 
728 
786 



9.99368 

9-99394 
9.99419 

9-99444 
9.99469 



901 
958 

99016 

073 



9-99495 
9.99520 

9-99545 
9.99570 
9.99596 



99i3i 
189 
247 

304 
362 



9.99621 
9.99646 
9.99672 
9.99697 
9.99722 



99420 
478 
536 
594 
652 



9-99747 
9-99773 
9.99798 
9.99823 
9.99848 



99710 
768 



942 

IOOOO 



9.99874 
9.99899 
9.99924 
9.99949 

9-99975 
0.00000 



0.01516 
0.01491 
0.01466 
0.01440 
0.01415 



I-035S 
349 
343 
337 
33i 



0.01390 
0.01365 
0.01339 
0.01314 
0.01289 



1.0325 
3i9 
3i3 
307 
301 



0.01263 
0.01238 
0.01213 
0.01188 
0.01162 



0.01137 
0.01112 
0.01087 
0.01061 
0.01036 



1.0295 
289 
283 
277 
271 

1.0265 
259 
253 
247 
241 



0.00985 
0.00960 
0.00935 
0.00910 



1.0235 
230 
224 
218 
212 



0.00884 
0.00859 
0.00834 
0.00809 
0.00783 



1.0206 
200 
194 
188 



0.00758 
0.00733 
0.00707 
0.00682 
0.00657 



1.0176 
170 
164 
158 
152 



0.00632 
0.00606 
0.00581 
0.00556 
O.QQ53 1 



1.0147 
141 

135 
129 
123 



0.00505 
0.00480 
0.00455 
0.00430 
0.00404 



1.0117 
in 
105 
099 
094 



0.00379 
0.00354 
0.00328 
0.00303 
0.00278 



1.0088 
082 
076 
070 



0.00253 
0.00227 
0.00202 
0.00177 
0.00152 



1.0058 
052 

047 
041 

035 



0.00126 

O.OOIOI 

0.00076 
0.00051 
0.00025 

0.00000 



1.0029 
023 
017 
012 
006 

000 



Nat. COS Log. d. Nat. Sin Log. d. Nat. CotLog. c.d. 



Log.TanNat. 



OCT 21 



/ 



